Theorem preqd ≪ | index | src | ≫

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


_G -> _a1 = _a2
_G -> _b1 = _b2
_G -> _a1 + _b1 = _a2 + _b2
_G -> suc (_a1 + _b1) = suc (_a2 + _b2)
_G -> (_a1 + _b1) * suc (_a1 + _b1) = (_a2 + _b2) * suc (_a2 + _b2)
_G -> 2 = 2
_G -> (_a1 + _b1) * suc (_a1 + _b1) // 2 = (_a2 + _b2) * suc (_a2 + _b2) // 2
_G -> (_a1 + _b1) * suc (_a1 + _b1) // 2 + _b1 = (_a2 + _b2) * suc (_a2 + _b2) // 2 + _b2
_G -> _a1, _b1 = _a2, _b2

Step	Hyp	Ref	Expression
1		hyp _ah	_G -> _a1 = _a2
2		hyp _bh	_G -> _b1 = _b2
3	1, 2	addeqd	_G -> _a1 + _b1 = _a2 + _b2
4	3	suceqd	_G -> suc (_a1 + _b1) = suc (_a2 + _b2)
5	3, 4	muleqd	_G -> (_a1 + _b1) * suc (_a1 + _b1) = (_a2 + _b2) * suc (_a2 + _b2)
6		eqidd	_G -> 2 = 2
7	5, 6	diveqd	_G -> (_a1 + _b1) * suc (_a1 + _b1) // 2 = (_a2 + _b2) * suc (_a2 + _b2) // 2
8	7, 2	addeqd	_G -> (_a1 + _b1) * suc (_a1 + _b1) // 2 + _b1 = (_a2 + _b2) * suc (_a2 + _b2) // 2 + _b2
9	8	conv pr	_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)