Theorem grpoinvid1 30461

Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinv.1	⊢ 𝑋 = ran 𝐺
grpinv.2	⊢ 𝑈 = (GId‘𝐺)
grpinv.3	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
grpoinvid1	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Proof of Theorem grpoinvid1

Step	Hyp	Ref	Expression
1		oveq2 7432	. . . 4 ⊢ ((𝑁‘𝐴) = 𝐵 → (𝐴𝐺(𝑁‘𝐴)) = (𝐴𝐺𝐵))
2	1	adantl 480	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐴𝐺(𝑁‘𝐴)) = (𝐴𝐺𝐵))
3		grpinv.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4		grpinv.2	. . . . . 6 ⊢ 𝑈 = (GId‘𝐺)
5		grpinv.3	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
6	3, 4, 5	grporinv 30460	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
7	6	3adant3 1129	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
8	7	adantr 479	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐴𝐺(𝑁‘𝐴)) = 𝑈)
9	2, 8	eqtr3d 2768	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑁‘𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10		oveq2 7432	. . . 4 ⊢ ((𝐴𝐺𝐵) = 𝑈 → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁‘𝐴)𝐺𝑈))
11	10	adantl 480	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁‘𝐴)𝐺𝑈))
12	3, 4, 5	grpolinv 30459	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈)
13	12	oveq1d 7439	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14	13	3adant3 1129	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
15	3, 5	grpoinvcl 30457	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
16	15	adantrr 715	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘𝐴) ∈ 𝑋)
17		simprl 769	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
18		simprr 771	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
19	16, 17, 18	3jca 1125	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
20	3	grpoass 30436	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑁‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
21	19, 20	syldan 589	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
22	21	3impb 1112	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
23	14, 22	eqtr3d 2768	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)))
24	3, 4	grpolid 30449	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
25	24	3adant2 1128	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵)
26	23, 25	eqtr3d 2768	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
27	26	adantr 479	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁‘𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
28	3, 4	grporid 30450	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
29	15, 28	syldan 589	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
30	29	3adant3 1129	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
31	30	adantr 479	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁‘𝐴)𝐺𝑈) = (𝑁‘𝐴))
32	11, 27, 31	3eqtr3rd 2775	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁‘𝐴) = 𝐵)
33	9, 32	impbida 799	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ran crn 5683  ‘cfv 6554  (class class class)co 7424  GrpOpcgr 30422  GIdcgi 30423  invcgn 30424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-grpo 30426  df-gid 30427  df-ginv 30428

This theorem is referenced by:  grpoinvop  30466  rngonegmn1l  37642