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Theorem grpodivdiv 30541

Description: Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
grpdivf.1	⊢ 𝑋 = ran 𝐺
grpdivf.3	⊢ 𝐷 = ( /_𝑔 ‘𝐺)

**Assertion**
Ref	Expression
grpodivdiv	⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵)))

**Proof of Theorem grpodivdiv**
Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
2		simpr1 1195	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
3		grpdivf.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
4		grpdivf.3	. . . . 5 ⊢ 𝐷 = ( /_𝑔 ‘𝐺)
5	3, 4	grpodivcl 30540	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) ∈ 𝑋)
6	5	3adant3r1 1183	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷𝐶) ∈ 𝑋)
7		eqid 2733	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
8	3, 7, 4	grpodivval 30536	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺((inv‘𝐺)‘(𝐵𝐷𝐶))))
9	1, 2, 6, 8	syl3anc 1373	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺((inv‘𝐺)‘(𝐵𝐷𝐶))))
10	3, 7, 4	grpoinvdiv 30538	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘(𝐵𝐷𝐶)) = (𝐶𝐷𝐵))
11	10	3adant3r1 1183	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘(𝐵𝐷𝐶)) = (𝐶𝐷𝐵))
12	11	oveq2d 7371	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺((inv‘𝐺)‘(𝐵𝐷𝐶))) = (𝐴𝐺(𝐶𝐷𝐵)))
13	9, 12	eqtrd 2768	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ran crn 5622 ‘cfv 6489 (class class class)co 7355 GrpOpcgr 30490 invcgn 30492 /_𝑔 cgs 30493

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 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-1st 7930 df-2nd 7931 df-grpo 30494 df-gid 30495 df-ginv 30496 df-gdiv 30497

This theorem is referenced by: ablodivdiv 30554

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