Theorem grpsubpropd2 13690

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 1023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝜑)
2		simp2 1024	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ (Base‘𝐺))
3		grpsubpropd2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
4	3	3ad2ant1 1044	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐵 = (Base‘𝐺))
5	2, 4	eleqtrrd 2311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑎 ∈ 𝐵)
6		grpsubpropd2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
7	6	3ad2ant1 1044	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
8		simp3 1025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → 𝑏 ∈ (Base‘𝐺))
9		eqid 2231	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺)
10		eqid 2231	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 10	grpinvcl 13633	. . . . . . . 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 2311	. . . . . 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 6045	. . . . . 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 1271	. . . . 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 2231	. . . . . . . . . . . . 13 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	9, 18	grpidcl 13614	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
20	6, 19	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐺) ∈ (Base‘𝐺))
21	20, 3	eleqtrrd 2311	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐵)
22	17, 21	basmexd 13145	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ V)
23	3, 17, 6, 22, 14	grpinvpropdg 13660	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
24	23	fveq1d 5641	. . . . . . 7 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
25	24	oveq2d 6034	. . . . . 6 ⊢ (𝜑 → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
26	25	3ad2ant1 1044	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐻)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
27	16, 26	eqtrd 2264	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
28	27	mpoeq3dva 6085	. . 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 2266	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
30		mpoeq12 6081	. . . 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 2264	. 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 2231	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
34		eqid 2231	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
35	9, 33, 10, 34	grpsubfvalg 13630	. . 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 2231	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
38		eqid 2231	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
39		eqid 2231	. . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻)
40		eqid 2231	. . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻)
41	37, 38, 39, 40	grpsubfvalg 13630	. . 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 2274	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020  Basecbs 13084  +_gcplusg 13162  0_gc0g 13341  Grpcgrp 13585  inv_gcminusg 13586  -_gcsg 13587

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-inn 9144  df-2 9202  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-0g 13343  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-grp 13588  df-minusg 13589  df-sbg 13590

This theorem is referenced by: (None)