Theorem isgrpinv 18935

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 18922	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑁‘𝑥) = (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ))
6	5	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑁‘𝑥) = (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ))
7		simpr 484	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → ((𝑀‘𝑥) + 𝑥) = 0 )
8		simpllr 776	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑀:𝐵⟶𝐵)
9		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑥 ∈ 𝐵)
10	8, 9	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑀‘𝑥) ∈ 𝐵)
11	1, 2, 3	grpinveu 18916	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 )
12	11	ad4ant13 752	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 )
13		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑀‘𝑥) → (𝑒 + 𝑥) = ((𝑀‘𝑥) + 𝑥))
14	13	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑀‘𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
15	14	riota2 7350	. . . . . . . . . 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 585	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀‘𝑥))
18	6, 17	eqtrd 2772	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑁‘𝑥) = (𝑀‘𝑥))
19	18	ex 412	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) → (((𝑀‘𝑥) + 𝑥) = 0 → (𝑁‘𝑥) = (𝑀‘𝑥)))
20	19	ralimdva 3150	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) → (∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
21	20	impr 454	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥))
22	1, 4	grpinvfn 18923	. . . . 5 ⊢ 𝑁 Fn 𝐵
23		ffn 6670	. . . . . 6 ⊢ (𝑀:𝐵⟶𝐵 → 𝑀 Fn 𝐵)
24	23	ad2antrl 729	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
25		eqfnfv 6985	. . . . 5 ⊢ ((𝑁 Fn 𝐵 ∧ 𝑀 Fn 𝐵) → (𝑁 = 𝑀 ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
26	22, 24, 25	sylancr 588	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → (𝑁 = 𝑀 ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
27	21, 26	mpbird 257	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → 𝑁 = 𝑀)
28	27	ex 412	. 2 ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑁 = 𝑀))
29	1, 4	grpinvf 18928	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
30	1, 2, 3, 4	grplinv 18931	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑥) + 𝑥) = 0 )
31	30	ralrimiva 3130	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 )
32	29, 31	jca 511	. . 3 ⊢ (𝐺 ∈ Grp → (𝑁:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ))
33		feq1 6648	. . . 4 ⊢ (𝑁 = 𝑀 → (𝑁:𝐵⟶𝐵 ↔ 𝑀:𝐵⟶𝐵))
34		fveq1 6841	. . . . . . 7 ⊢ (𝑁 = 𝑀 → (𝑁‘𝑥) = (𝑀‘𝑥))
35	34	oveq1d 7383	. . . . . 6 ⊢ (𝑁 = 𝑀 → ((𝑁‘𝑥) + 𝑥) = ((𝑀‘𝑥) + 𝑥))
36	35	eqeq1d 2739	. . . . 5 ⊢ (𝑁 = 𝑀 → (((𝑁‘𝑥) + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
37	36	ralbidv 3161	. . . 4 ⊢ (𝑁 = 𝑀 → (∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ))
38	33, 37	anbi12d 633	. . 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 245	. 2 ⊢ (𝐺 ∈ Grp → (𝑁 = 𝑀 → (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )))
40	28, 39	impbid 212	1 ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3350   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Grpcgrp 18875  inv_gcminusg 18876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-riota 7325  df-ov 7371  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879

This theorem is referenced by:  oppginv  19300