Theorem grpsubpropdg 12833

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 2176	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2176	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		grpsubpropdg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6		grpsubpropdg.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
7	2	oveqdr 5893	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
8	4, 1, 5, 6, 7	grpinvpropdg 12804	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
9	8	fveq1d 5509	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
10	2, 3, 9	oveq123d 5886	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
11	1, 1, 10	mpoeq123dv 5927	. 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 2175	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2175	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2175	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
15		eqid 2175	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
16	12, 13, 14, 15	grpsubfvalg 12778	. . 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 2175	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2175	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
20		eqid 2175	. . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻)
21		eqid 2175	. . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻)
22	18, 19, 20, 21	grpsubfvalg 12778	. . 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 2218	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2146  ‘cfv 5208  (class class class)co 5865   ∈ cmpo 5867  Basecbs 12428  +_gcplusg 12492  inv_gcminusg 12739  -_gcsg 12740

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-inn 8891  df-ndx 12431  df-slot 12432  df-base 12434  df-0g 12628  df-minusg 12742  df-sbg 12743

This theorem is referenced by: (None)