Theorem grpsubpropdg 12860

Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)

Hypotheses

Ref	Expression
grpsubpropd.b	⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
grpsubpropd.p	⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))
grpsubpropdg.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
grpsubpropdg.h	⊢ (𝜑 → 𝐻 ∈ 𝑊)

Assertion

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

Proof of Theorem grpsubpropdg

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

Step	Hyp	Ref	Expression
1		grpsubpropd.b	. . 3 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
2		grpsubpropd.p	. . . 4 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))
3		eqidd 2178	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2178	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		grpsubpropdg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6		grpsubpropdg.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
7	2	oveqdr 5897	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
8	4, 1, 5, 6, 7	grpinvpropdg 12831	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
9	8	fveq1d 5513	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
10	2, 3, 9	oveq123d 5890	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
11	1, 1, 10	mpoeq123dv 5931	. 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))))
12		eqid 2177	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2177	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2177	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
15		eqid 2177	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
16	12, 13, 14, 15	grpsubfvalg 12805	. . 3 ⊢ (𝐺 ∈ 𝑉 → (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))))
17	5, 16	syl 14	. 2 ⊢ (𝜑 → (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))))
18		eqid 2177	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2177	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
20		eqid 2177	. . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻)
21		eqid 2177	. . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻)
22	18, 19, 20, 21	grpsubfvalg 12805	. . 3 ⊢ (𝐻 ∈ 𝑊 → (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
23	6, 22	syl 14	. 2 ⊢ (𝜑 → (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))))
24	11, 17, 23	3eqtr4d 2220	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ‘cfv 5212  (class class class)co 5869   ∈ cmpo 5871  Basecbs 12442  +_gcplusg 12515  inv_gcminusg 12765  -_gcsg 12766

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-inn 8906  df-ndx 12445  df-slot 12446  df-base 12448  df-0g 12652  df-minusg 12768  df-sbg 12769

This theorem is referenced by: (None)