Theorem grpsubpropdg 13901

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 2235	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2235	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		grpsubpropdg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6		grpsubpropdg.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
7	2	oveqdr 6086	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
8	4, 1, 5, 6, 7	grpinvpropdg 13872	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
9	8	fveq1d 5677	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
10	2, 3, 9	oveq123d 6079	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
11	1, 1, 10	mpoeq123dv 6123	. 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 2234	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2234	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2234	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
15		eqid 2234	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
16	12, 13, 14, 15	grpsubfvalg 13842	. . 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 2234	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2234	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
20		eqid 2234	. . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻)
21		eqid 2234	. . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻)
22	18, 19, 20, 21	grpsubfvalg 13842	. . 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 2277	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060  Basecbs 13296  +_gcplusg 13374  inv_gcminusg 13798  -_gcsg 13799

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555  df-minusg 13801  df-sbg 13802

This theorem is referenced by:  aprprop  14524  rlmsubg  14718