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Theorem grpodivinv 30502

Description: Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
grpdiv.1	⊢ 𝑋 = ran 𝐺
grpdiv.2	⊢ 𝑁 = (inv‘𝐺)
grpdiv.3	⊢ 𝐷 = ( /_𝑔 ‘𝐺)

**Assertion**
Ref	Expression
grpodivinv	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺𝐵))

**Proof of Theorem grpodivinv**
Step	Hyp	Ref	Expression
1		grpdiv.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		grpdiv.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
3	1, 2	grpoinvcl 30490	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
4	3	3adant2 1131	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
5		grpdiv.3	. . . 4 ⊢ 𝐷 = ( /_𝑔 ‘𝐺)
6	1, 2, 5	grpodivval 30501	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐵) ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺(𝑁‘(𝑁‘𝐵))))
7	4, 6	syld3an3 1410	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺(𝑁‘(𝑁‘𝐵))))
8	1, 2	grpo2inv 30497	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
9	8	3adant2 1131	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
10	9	oveq2d 7430	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘(𝑁‘𝐵))) = (𝐴𝐺𝐵))
11	7, 10	eqtrd 2769	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ran crn 5668 ‘cfv 6542 (class class class)co 7414 GrpOpcgr 30455 invcgn 30457 /_𝑔 cgs 30458

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-id 5560 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7997 df-2nd 7998 df-grpo 30459 df-gid 30460 df-ginv 30461 df-gdiv 30462

This theorem is referenced by: ablodivdiv4 30520

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