Theorem grpoidinvlem2 30491

Description: Lemma for grpoidinv 30494. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpoidinvlem2	⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))

Proof of Theorem grpoidinvlem2

Step	Hyp	Ref	Expression
1		simprr 772	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		simprl 770	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝑌 ∈ 𝑋)
3		grpfo.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
4	3	grpocl 30486	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
5	4	3com23 1126	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
6	5	3expb 1120	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
7	1, 2, 6	3jca 1128	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
8	3	grpoass 30489	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
9	7, 8	syldan 591	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
10	9	adantr 480	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11		oveq1 7417	. . . . . . 7 ⊢ ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
12	11	adantl 481	. . . . . 6 ⊢ (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13		simpl 482	. . . . . 6 ⊢ (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
14	12, 13	eqtr2d 2772	. . . . 5 ⊢ (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → 𝑌 = ((𝑌𝐺𝐴)𝐺𝑌))
15		id 22	. . . . . . 7 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋))
16	15	3anidm13 1422	. . . . . 6 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋))
17	3	grpoass 30489	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
18	16, 17	sylan2 593	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
19	14, 18	sylan9eqr 2793	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
20	19	eqcomd 2742	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
21	20	oveq2d 7426	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
22	10, 21	eqtrd 2771	1 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ran crn 5660  (class class class)co 7410  GrpOpcgr 30475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7413  df-grpo 30479

This theorem is referenced by:  grpoidinvlem3  30492