Theorem isgrpinv 19184

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 19171	. . . . . . . . 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 7077	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑀‘𝑥) ∈ 𝐵)
11	1, 2, 3	grpinveu 19165	. . . . . . . . . . 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 7419	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑀‘𝑥) → (𝑒 + 𝑥) = ((𝑀‘𝑥) + 𝑥))
14	13	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑀‘𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
15	14	riota2 7394	. . . . . . . . . 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 2796	. . . . . . 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 3175	. . . . 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 19172	. . . . 5 ⊢ 𝑁 Fn 𝐵
23		ffn 6701	. . . . . 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 7021	. . . . 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 19177	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
30	1, 2, 3, 4	grplinv 19180	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑥) + 𝑥) = 0 )
31	30	ralrimiva 3155	. . . 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 6679	. . . 4 ⊢ (𝑁 = 𝑀 → (𝑁:𝐵⟶𝐵 ↔ 𝑀:𝐵⟶𝐵))
34		fveq1 6876	. . . . . . 7 ⊢ (𝑁 = 𝑀 → (𝑁‘𝑥) = (𝑀‘𝑥))
35	34	oveq1d 7427	. . . . . 6 ⊢ (𝑁 = 𝑀 → ((𝑁‘𝑥) + 𝑥) = ((𝑀‘𝑥) + 𝑥))
36	35	eqeq1d 2763	. . . . 5 ⊢ (𝑁 = 𝑀 → (((𝑁‘𝑥) + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
37	36	ralbidv 3186	. . . 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 2145  ∀wral 3077  ∃!wreu 3364   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  ℩crio 7368  (class class class)co 7412  Basecbs 17367  +_gcplusg 17408  0_gc0g 17590  Grpcgrp 19124  inv_gcminusg 19125

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  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 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-riota 7369  df-ov 7415  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-minusg 19128

This theorem is used by:  oppginv  19553