Theorem grpoidinvlem1 29752

Description: Lemma for grpoidinv 29756. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpoidinvlem1	⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈)

Proof of Theorem grpoidinvlem1

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
2	1	3anidm23 1421	. . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
3		grpfo.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	3	grpoass 29751	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴)))
5	2, 4	sylan2 593	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴)))
6	5	adantr 481	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴)))
7		oveq1 7415	. . 3 ⊢ ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑈𝐺𝐴))
8	7	ad2antrl 726	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑈𝐺𝐴))
9		oveq2 7416	. . . 4 ⊢ ((𝐴𝐺𝐴) = 𝐴 → (𝑌𝐺(𝐴𝐺𝐴)) = (𝑌𝐺𝐴))
10	9	ad2antll 727	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺(𝐴𝐺𝐴)) = (𝑌𝐺𝐴))
11		simprl 769	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺𝐴) = 𝑈)
12	10, 11	eqtrd 2772	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺(𝐴𝐺𝐴)) = 𝑈)
13	6, 8, 12	3eqtr3d 2780	1 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5677  (class class class)co 7408  GrpOpcgr 29737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7411  df-grpo 29741

This theorem is referenced by:  grpoidinvlem3  29754