Theorem grpsubpropdg 13969

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 2239	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2239	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		grpsubpropdg.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6		grpsubpropdg.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
7	2	oveqdr 6116	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
8	4, 1, 5, 6, 7	grpinvpropdg 13940	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
9	8	fveq1d 5700	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
10	2, 3, 9	oveq123d 6109	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
11	1, 1, 10	mpoeq123dv 6153	. 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 2238	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2238	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2238	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
15		eqid 2238	. . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺)
16	12, 13, 14, 15	grpsubfvalg 13910	. . 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 2238	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2238	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
20		eqid 2238	. . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻)
21		eqid 2238	. . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻)
22	18, 19, 20, 21	grpsubfvalg 13910	. . 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 2281	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ‘cfv 5380  (class class class)co 6088   ∈ cmpo 6090  Basecbs 13411  +_gcplusg 13491  inv_gcminusg 13866  -_gcsg 13867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8277  ax-resscn 8278  ax-1re 8280  ax-addrcl 8283

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-inn 9315  df-ndx 13414  df-slot 13415  df-base 13417  df-0g 13672  df-minusg 13869  df-sbg 13870

This theorem is used by:  aprprop  14711  rlmsubg  14905