Theorem grpoidinvlem2 30664

Description: Lemma for grpoidinv 30667. (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 782	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		simprl 780	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝑌 ∈ 𝑋)
3		grpfo.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
4	3	grpocl 30659	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
5	4	3com23 1138	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
6	5	3expb 1132	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
7	1, 2, 6	3jca 1140	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
8	3	grpoass 30662	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
9	7, 8	syldan 600	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
10	9	adantr 484	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11		oveq1 7397	. . . . . . 7 ⊢ ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
12	11	adantl 485	. . . . . 6 ⊢ (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13		simpl 486	. . . . . 6 ⊢ (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
14	12, 13	eqtr2d 2797	. . . . 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 1438	. . . . . 6 ⊢ ((𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋))
17	3	grpoass 30662	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
18	16, 17	sylan2 602	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
19	14, 18	sylan9eqr 2818	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
20	19	eqcomd 2767	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
21	20	oveq2d 7406	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
22	10, 21	eqtrd 2796	1 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑌 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ran crn 5644  (class class class)co 7390  GrpOpcgr 30648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fo 6521  df-fv 6523  df-ov 7393  df-grpo 30652

This theorem is referenced by:  grpoidinvlem3  30665