Theorem isgrpinv 13256

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 13245	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (𝑁‘𝑥) = (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ))
6	5	ad2antlr 489	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑁‘𝑥) = (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ))
7		simpr 110	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → ((𝑀‘𝑥) + 𝑥) = 0 )
8		simpllr 534	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑀:𝐵⟶𝐵)
9		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑥 ∈ 𝐵)
10	8, 9	ffvelcdmd 5701	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑀‘𝑥) ∈ 𝐵)
11	1, 2, 3	grpinveu 13240	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 )
12	11	ad4ant13 513	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → ∃!𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 )
13		oveq1 5932	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑀‘𝑥) → (𝑒 + 𝑥) = ((𝑀‘𝑥) + 𝑥))
14	13	eqeq1d 2205	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑀‘𝑥) → ((𝑒 + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
15	14	riota2 5903	. . . . . . . . . 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 411	. . . . . . . . 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 147	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (℩𝑒 ∈ 𝐵 (𝑒 + 𝑥) = 0 ) = (𝑀‘𝑥))
18	6, 17	eqtrd 2229	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) ∧ ((𝑀‘𝑥) + 𝑥) = 0 ) → (𝑁‘𝑥) = (𝑀‘𝑥))
19	18	ex 115	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) ∧ 𝑥 ∈ 𝐵) → (((𝑀‘𝑥) + 𝑥) = 0 → (𝑁‘𝑥) = (𝑀‘𝑥)))
20	19	ralimdva 2564	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑀:𝐵⟶𝐵) → (∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
21	20	impr 379	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥))
22	1, 4	grpinvfng 13246	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝑁 Fn 𝐵)
23		ffn 5410	. . . . . 6 ⊢ (𝑀:𝐵⟶𝐵 → 𝑀 Fn 𝐵)
24	23	ad2antrl 490	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → 𝑀 Fn 𝐵)
25		eqfnfv 5662	. . . . 5 ⊢ ((𝑁 Fn 𝐵 ∧ 𝑀 Fn 𝐵) → (𝑁 = 𝑀 ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
26	22, 24, 25	syl2an2r 595	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → (𝑁 = 𝑀 ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) = (𝑀‘𝑥)))
27	21, 26	mpbird 167	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )) → 𝑁 = 𝑀)
28	27	ex 115	. 2 ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) → 𝑁 = 𝑀))
29	1, 4	grpinvf 13249	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
30	1, 2, 3, 4	grplinv 13252	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑥) + 𝑥) = 0 )
31	30	ralrimiva 2570	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 )
32	29, 31	jca 306	. . 3 ⊢ (𝐺 ∈ Grp → (𝑁:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ))
33		feq1 5393	. . . 4 ⊢ (𝑁 = 𝑀 → (𝑁:𝐵⟶𝐵 ↔ 𝑀:𝐵⟶𝐵))
34		fveq1 5560	. . . . . . 7 ⊢ (𝑁 = 𝑀 → (𝑁‘𝑥) = (𝑀‘𝑥))
35	34	oveq1d 5940	. . . . . 6 ⊢ (𝑁 = 𝑀 → ((𝑁‘𝑥) + 𝑥) = ((𝑀‘𝑥) + 𝑥))
36	35	eqeq1d 2205	. . . . 5 ⊢ (𝑁 = 𝑀 → (((𝑁‘𝑥) + 𝑥) = 0 ↔ ((𝑀‘𝑥) + 𝑥) = 0 ))
37	36	ralbidv 2497	. . . 4 ⊢ (𝑁 = 𝑀 → (∀𝑥 ∈ 𝐵 ((𝑁‘𝑥) + 𝑥) = 0 ↔ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ))
38	33, 37	anbi12d 473	. . 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 155	. 2 ⊢ (𝐺 ∈ Grp → (𝑁 = 𝑀 → (𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 )))
40	28, 39	impbid 129	1 ⊢ (𝐺 ∈ Grp → ((𝑀:𝐵⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑀‘𝑥) + 𝑥) = 0 ) ↔ 𝑁 = 𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃!wreu 2477   Fn wfn 5254  ⟶wf 5255  ‘cfv 5259  ℩crio 5879  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  Grpcgrp 13202  inv_gcminusg 13203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206

This theorem is referenced by: (None)