Theorem diveqd ≪ | index | src | ≫

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

Step	Hyp	Ref	Expression
1		eqidd	_G -> r = r
2		hyp _bh	_G -> _b1 = _b2
3	1, 2	lteqd	_G -> (r < _b1 <-> r < _b2)
4		eqidd	_G -> q = q
5	2, 4	muleqd	_G -> _b1 * q = _b2 * q
6	5, 1	addeqd	_G -> _b1 * q + r = _b2 * q + r
7		hyp _ah	_G -> _a1 = _a2
8	6, 7	eqeqd	_G -> (_b1 * q + r = _a1 <-> _b2 * q + r = _a2)
9	3, 8	aneqd	_G -> (r < _b1 /\ _b1 * q + r = _a1 <-> r < _b2 /\ _b2 * q + r = _a2)
10	9	exeqd	_G -> (E. r (r < _b1 /\ _b1 * q + r = _a1) <-> E. r (r < _b2 /\ _b2 * q + r = _a2))
11	10	abeqd	_G -> {q \| E. r (r < _b1 /\ _b1 * q + r = _a1)} == {q \| E. r (r < _b2 /\ _b2 * q + r = _a2)}
12	11	theeqd	_G -> the {q \| E. r (r < _b1 /\ _b1 * q + r = _a1)} = the {q \| E. r (r < _b2 /\ _b2 * q + r = _a2)}
13	12	conv div	_G -> _a1 // _b1 = _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 (peano2, addeq, muleq)