Theorem grpsubpropd 18975

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))

Assertion

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

Proof of Theorem grpsubpropd

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 2737	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5	2	oveqdr 7386	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
6	4, 1, 5	grpinvpropd 18945	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
7	6	fveq1d 6836	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
8	2, 3, 7	oveq123d 7379	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
9	1, 1, 8	mpoeq123dv 7433	. 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))))
10		eqid 2736	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2736	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
12		eqid 2736	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
13		eqid 2736	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
14	10, 11, 12, 13	grpsubfval 18913	. 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))
15		eqid 2736	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
16		eqid 2736	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
17		eqid 2736	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
18		eqid 2736	. . 3 ⊢ (-g‘𝐻) = (-g‘𝐻)
19	15, 16, 17, 18	grpsubfval 18913	. 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
20	9, 14, 19	3eqtr4g 2796	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17136  +_gcplusg 17177  inv_gcminusg 18864  -_gcsg 18865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-0g 17361  df-minusg 18867  df-sbg 18868

This theorem is referenced by:  rlmsub  21148  matsubg  22376  tngngp2  24596  tngngp  24598  tcphsub  25177  ply1divalg2  26100  ttgsub  28951  zhmnrg  34122