Theorem bgcdeqd ≪ | index | src | ≫

theorem bgcdeqd (_G: wff) (_a1 _a2 _b1 _b2: nat):
  $ _G -> _a1 = _a2 $ >
  $ _G -> _b1 = _b2 $ >
  $ _G -> bgcd _a1 _b1 = bgcd _a2 _b2 $;

Step	Hyp	Ref	Expression
1		biidd	_G -> (0 < d <-> 0 < d)
2		eqidd	_G -> x = x
3		hyp _ah	_G -> _a1 = _a2
4	2, 3	muleqd	_G -> x * _a1 = x * _a2
5		eqidd	_G -> y = y
6		hyp _bh	_G -> _b1 = _b2
7	5, 6	muleqd	_G -> y * _b1 = y * _b2
8		eqidd	_G -> d = d
9	7, 8	addeqd	_G -> y * _b1 + d = y * _b2 + d
10	4, 9	eqeqd	_G -> (x * _a1 = y * _b1 + d <-> x * _a2 = y * _b2 + d)
11	10	exeqd	_G -> (E. y x * _a1 = y * _b1 + d <-> E. y x * _a2 = y * _b2 + d)
12	11	exeqd	_G -> (E. x E. y x * _a1 = y * _b1 + d <-> E. x E. y x * _a2 = y * _b2 + d)
13	1, 12	aneqd	_G -> (0 < d /\ E. x E. y x * _a1 = y * _b1 + d <-> 0 < d /\ E. x E. y x * _a2 = y * _b2 + d)
14	13	abeqd	_G -> {d \| 0 < d /\ E. x E. y x * _a1 = y * _b1 + d} == {d \| 0 < d /\ E. x E. y x * _a2 = y * _b2 + d}
15	14	leasteqd	_G -> least {d \| 0 < d /\ E. x E. y x * _a1 = y * _b1 + d} = least {d \| 0 < d /\ E. x E. y x * _a2 = y * _b2 + d}
16	15	conv bgcd	_G -> bgcd _a1 _b1 = bgcd _a2 _b2

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (addeq, muleq)