Theorem isgrpinv 19233

Description: Properties showing that a function 𝑀 is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
grpinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinv.p	⊢ + = (+g‘𝐺)
grpinv.u	⊢ 0 = (0g‘𝐺)
grpinv.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
isgrpinv	⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥, 0   𝑥, +   𝑥,𝑀   𝑥,𝑁

Proof of Theorem isgrpinv

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinv.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
2		grpinv.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
3		grpinv.u	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐺)
4		grpinv.n	. . . . . . . . . 10 ⊢ 𝑁 = (invg‘𝐺)
5	1, 2, 3, 4	grpinvval 19220	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑁‘𝑥) = (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ))
6	5	ad2antlr 740	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑁‘𝑥) = (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ))
7		simpr 490	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → ((𝑀‘𝑥) + 𝑥) = 0 )
8		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑀:𝐵⟶𝐵)
9		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑥 ∈ 𝐵)
10	8, 9	ffvelcdmd 7109	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑀‘𝑥) ∈ 𝐵)
11	1, 2, 3	grpinveu 19214	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 )
12	11	ad4ant13 764	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 )
13		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑀‘𝑥) → (𝑒 + 𝑥) = ((𝑀‘𝑥) + 𝑥))
14	13	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑀‘𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
15	14	riota2 7425	. . . . . . . . . 10 ⊢ (((𝑀‘𝑥) ∈ 𝐵 ∧ ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ) → (((𝑀‘𝑥) + 𝑥) = 0 ↔ (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀‘𝑥)))
16	10, 12, 15	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (((𝑀‘𝑥) + 𝑥) = 0 ↔ (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀‘𝑥)))
17	7, 16	mpbid 235	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀‘𝑥))
18	6, 17	eqtrd 2798	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑁‘𝑥) = (𝑀‘𝑥))
19	18	ex 418	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) → (((𝑀‘𝑥) + 𝑥) = 0 → (𝑁‘𝑥) = (𝑀‘𝑥)))
20	19	ralimdva 3178	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) → (∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
21	20	impr 460	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥))
22	1, 4	grpinvfn 19221	. . . . 5 ⊢ 𝑁 Fn 𝐵
23		ffn 6731	. . . . . 6 ⊢ (𝑀:𝐵⟶𝐵 → 𝑀 Fn 𝐵)
24	23	ad2antrl 741	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
25		eqfnfv 7052	. . . . 5 ⊢ ((𝑁 Fn 𝐵 ∧ 𝑀 Fn 𝐵) → (𝑁 = 𝑀 ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
26	22, 24, 25	sylancr 599	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → (𝑁 = 𝑀 ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
27	21, 26	mpbird 260	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → 𝑁 = 𝑀)
28	27	ex 418	. 2 ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑁 = 𝑀))
29	1, 4	grpinvf 19226	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
30	1, 2, 3, 4	grplinv 19229	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑥) + 𝑥) = 0 )
31	30	ralrimiva 3158	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 )
32	29, 31	jca 521	. . 3 ⊢ (𝐺 ∈ Grp → (𝑁:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ))
33		feq1 6709	. . . 4 ⊢ (𝑁 = 𝑀 → (𝑁:𝐵⟶𝐵 ↔ 𝑀:𝐵⟶𝐵))
34		fveq1 6907	. . . . . . 7 ⊢ (𝑁 = 𝑀 → (𝑁‘𝑥) = (𝑀‘𝑥))
35	34	oveq1d 7458	. . . . . 6 ⊢ (𝑁 = 𝑀 → ((𝑁‘𝑥) + 𝑥) = ((𝑀‘𝑥) + 𝑥))
36	35	eqeq1d 2765	. . . . 5 ⊢ (𝑁 = 𝑀 → (((𝑁‘𝑥) + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
37	36	ralbidv 3189	. . . 4 ⊢ (𝑁 = 𝑀 → (∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ))
38	33, 37	anbi12d 644	. . 3 ⊢ (𝑁 = 𝑀 → ((𝑁:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ) ↔ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )))
39	32, 38	syl5ibcom 248	. 2 ⊢ (𝐺 ∈ Grp → (𝑁 = 𝑀 → (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )))
40	28, 39	impbid 215	1 ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  ℩crio 7399  (class class class)co 7443  Basecbs 17416  +_gcplusg 17457  0_gc0g 17639  Grpcgrp 19173  inv_gcminusg 19174

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-riota 7400  df-ov 7446  df-0g 17641  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-grp 19176  df-minusg 19177

This theorem is used by:  oppginv  19603