Theorem grpsubpropd2 13307

Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
grpsubpropd2.1	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
grpsubpropd2.2	⊢ (𝜑 → 𝐵 = (Base‘𝐻))
grpsubpropd2.3	⊢ (𝜑 → 𝐺 ∈ Grp)
grpsubpropd2.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))

Assertion

Ref	Expression
grpsubpropd2	⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝜑,𝑥,𝑦

Proof of Theorem grpsubpropd2

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 999	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2		simp2 1000	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3		grpsubpropd2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
4	3	3ad2ant1 1020	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
5	2, 4	eleqtrrd 2276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ 𝐵)
6		grpsubpropd2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	3ad2ant1 1020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8		simp3 1001	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9		eqid 2196	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2196	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 13250	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
12	7, 8, 11	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
13	12, 4	eleqtrrd 2276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ 𝐵)
14		grpsubpropd2.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
15	14	oveqrspc2v 5952	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑏) ∈ 𝐵)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)))
16	1, 5, 13, 15	syl12anc 1247	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)))
17		grpsubpropd2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝐻))
18		eqid 2196	. . . . . . . . . . . . 13 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	9, 18	grpidcl 13231	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
20	6, 19	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐺) ∈ (Base‘𝐺))
21	20, 3	eleqtrrd 2276	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐵)
22	17, 21	basmexd 12763	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ V)
23	3, 17, 6, 22, 14	grpinvpropdg 13277	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
24	23	fveq1d 5563	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
25	24	oveq2d 5941	. . . . . 6 ⊢ (𝜑 → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
26	25	3ad2ant1 1020	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
27	16, 26	eqtrd 2229	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
28	27	mpoeq3dva 5990	. . 3 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
29	3, 17	eqtr3d 2231	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
30		mpoeq12 5986	. . . 4 ⊢ (((Base‘𝐺) = (Base‘𝐻) ∧ (Base‘𝐺) = (Base‘𝐻)) → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
31	29, 29, 30	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
32	28, 31	eqtrd 2229	. 2 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
33		eqid 2196	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
34		eqid 2196	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
35	9, 33, 10, 34	grpsubfvalg 13247	. . 3 ⊢ (𝐺 ∈ Grp → (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))))
36	6, 35	syl 14	. 2 ⊢ (𝜑 → (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))))
37		eqid 2196	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
38		eqid 2196	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
39		eqid 2196	. . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻)
40		eqid 2196	. . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻)
41	37, 38, 39, 40	grpsubfvalg 13247	. . 3 ⊢ (𝐻 ∈ V → (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
42	22, 41	syl 14	. 2 ⊢ (𝜑 → (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
43	32, 36, 42	3eqtr4d 2239	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  Vcvv 2763  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  Grpcgrp 13202  inv_gcminusg 13203  -_gcsg 13204

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-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  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  df-sbg 13207

This theorem is referenced by: (None)