Theorem grpsubpropd2 19077

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 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2		simp2 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3		grpsubpropd2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
4	3	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
5	2, 4	eleqtrrd 2842	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ 𝐵)
6		grpsubpropd2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8		simp3 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2735	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 19018	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
12	7, 8, 11	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
13	12, 4	eleqtrrd 2842	. . . . . 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 7458	. . . . . 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 837	. . . . 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 19046	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
19	18	fveq1d 6909	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
20	19	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
21	20	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
22	16, 21	eqtrd 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
23	22	mpoeq3dva 7510	. . 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 2777	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
25		mpoeq12 7506	. . . 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 584	. . 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 2775	. 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 2735	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
29		eqid 2735	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
30	9, 28, 10, 29	grpsubfval 19014	. 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))
31		eqid 2735	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
32		eqid 2735	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
33		eqid 2735	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
34		eqid 2735	. . 3 ⊢ (-g‘𝐻) = (-g‘𝐻)
35	31, 32, 33, 34	grpsubfval 19014	. 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
36	27, 30, 35	3eqtr4g 2800	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  +_gcplusg 17298  Grpcgrp 18964  inv_gcminusg 18965  -_gcsg 18966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969

This theorem is referenced by:  ngppropd  24666