Theorem grpsubpropd2 18469

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 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2		simp2 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3		grpsubpropd2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
4	3	3ad2ant1 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
5	2, 4	eleqtrrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ 𝐵)
6		grpsubpropd2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	3ad2ant1 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8		simp3 1140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2737	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 18415	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
12	7, 8, 11	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑏) ∈ (Base‘𝐺))
13	12, 4	eleqtrrd 2841	. . . . . 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 7240	. . . . . 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 18438	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
19	18	fveq1d 6719	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
20	19	oveq2d 7229	. . . . . 6 ⊢ (𝜑 → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
21	20	3ad2ant1 1135	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
22	16, 21	eqtrd 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
23	22	mpoeq3dva 7288	. . 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 2779	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
25		mpoeq12 7284	. . . 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 587	. . 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 2777	. 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 2737	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
29		eqid 2737	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
30	9, 28, 10, 29	grpsubfval 18411	. 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))
31		eqid 2737	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
32		eqid 2737	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
33		eqid 2737	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
34		eqid 2737	. . 3 ⊢ (-g‘𝐻) = (-g‘𝐻)
35	31, 32, 33, 34	grpsubfval 18411	. 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
36	27, 30, 35	3eqtr4g 2803	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  Basecbs 16760  +_gcplusg 16802  Grpcgrp 18365  inv_gcminusg 18366  -_gcsg 18367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370

This theorem is referenced by:  ngppropd  23535