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Theorem grpodivinv 31147

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 31135	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
4	3	3adant2 1149	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋)
5		grpdiv.3	. . . 4 ⊢ 𝐷 = ( /_𝑔 ‘𝐺)
6	1, 2, 5	grpodivval 31146	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐵) ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺(𝑁‘(𝑁‘𝐵))))
7	4, 6	syld3an3 1436	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺(𝑁‘(𝑁‘𝐵))))
8	1, 2	grpo2inv 31142	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
9	8	3adant2 1149	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵)
10	9	oveq2d 7440	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘(𝑁‘𝐵))) = (𝐴𝐺𝐵))
11	7, 10	eqtrd 2795	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝑁‘𝐵)) = (𝐴𝐺𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ran crn 5652 ‘cfv 6543 (class class class)co 7424 GrpOpcgr 31100 invcgn 31102 /_𝑔 cgs 31103

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8005 df-2nd 8006 df-grpo 31104 df-gid 31105 df-ginv 31106 df-gdiv 31107

This theorem is used by: ablodivdiv4 31165

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