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		hyp _ah	_G -> _a1 = _a2
2	1	muleq2d	_G -> x * _a1 = x * _a2
3		hyp _bh	_G -> _b1 = _b2
4	3	muleq2d	_G -> y * _b1 = y * _b2
5	4	addeq1d	_G -> y * _b1 + d = y * _b2 + d
6	2, 5	eqeqd	_G -> (x * _a1 = y * _b1 + d <-> x * _a2 = y * _b2 + d)
7	6	exeqd	_G -> (E. y x * _a1 = y * _b1 + d <-> E. y x * _a2 = y * _b2 + d)
8	7	exeqd	_G -> (E. x E. y x * _a1 = y * _b1 + d <-> E. x E. y x * _a2 = y * _b2 + d)
9	8	aneq2d	_G -> (0 < d /\ E. x E. y x * _a1 = y * _b1 + d <-> 0 < d /\ E. x E. y x * _a2 = y * _b2 + d)
10	9	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}
11	10	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}
12	11	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)