Theorem grpsubpropd2 18208

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 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2		simp2 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3		grpsubpropd2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
4	3	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
5	2, 4	eleqtrrd 2919	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ 𝐵)
6		grpsubpropd2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8		simp3 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9		eqid 2824	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2824	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 18154	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
12	7, 8, 11	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
13	12, 4	eleqtrrd 2919	. . . . . 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 7186	. . . . . 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 834	. . . . 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	3, 17, 14	grpinvpropd 18177	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
19	18	fveq1d 6675	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
20	19	oveq2d 7175	. . . . . 6 ⊢ (𝜑 → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
21	20	3ad2ant1 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
22	16, 21	eqtrd 2859	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
23	22	mpoeq3dva 7234	. . 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))))
24	3, 17	eqtr3d 2861	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
25		mpoeq12 7230	. . . 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))))
26	24, 24, 25	syl2anc 586	. . 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))))
27	23, 26	eqtrd 2859	. 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))))
28		eqid 2824	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
29		eqid 2824	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
30	9, 28, 10, 29	grpsubfval 18150	. 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))
31		eqid 2824	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
32		eqid 2824	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
33		eqid 2824	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
34		eqid 2824	. . 3 ⊢ (-g‘𝐻) = (-g‘𝐻)
35	31, 32, 33, 34	grpsubfval 18150	. 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
36	27, 30, 35	3eqtr4g 2884	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ‘cfv 6358  (class class class)co 7159   ∈ cmpo 7161  Basecbs 16486  +_gcplusg 16568  Grpcgrp 18106  inv_gcminusg 18107  -_gcsg 18108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-0g 16718  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-grp 18109  df-minusg 18110  df-sbg 18111

This theorem is referenced by:  ngppropd  23249