Theorem grpoidinvlem4 30582

Description: Lemma for grpoidinv 30583. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpoidinvlem4	⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋   𝑦,𝑈

Proof of Theorem grpoidinvlem4

Step	Hyp	Ref	Expression
1		simpll 766	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simplr 768	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simpr 484	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
4		grpfo.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	4	grpoass 30578	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
6	1, 2, 3, 2, 5	syl13anc 1374	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
7		oveq2 7366	. . . . 5 ⊢ ((𝑦𝐺𝐴) = 𝑈 → (𝐴𝐺(𝑦𝐺𝐴)) = (𝐴𝐺𝑈))
8	6, 7	sylan9eq 2791	. . . 4 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺𝑈))
9		oveq1 7365	. . . 4 ⊢ ((𝐴𝐺𝑦) = 𝑈 → ((𝐴𝐺𝑦)𝐺𝐴) = (𝑈𝐺𝐴))
10	8, 9	sylan9req 2792	. . 3 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) ∧ (𝐴𝐺𝑦) = 𝑈) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
11	10	anasss 466	. 2 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
12	11	r19.29an 3140	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ran crn 5625  (class class class)co 7358  GrpOpcgr 30564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-ov 7361  df-grpo 30568

This theorem is referenced by:  grpoidinv  30583  grpoideu  30584