Theorem grpsubpropd 18955

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 2732	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2732	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5	2	oveqdr 7374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
6	4, 1, 5	grpinvpropd 18925	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
7	6	fveq1d 6824	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
8	2, 3, 7	oveq123d 7367	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
9	1, 1, 8	mpoeq123dv 7421	. 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 2731	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2731	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
12		eqid 2731	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
13		eqid 2731	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
14	10, 11, 12, 13	grpsubfval 18893	. 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))
15		eqid 2731	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
16		eqid 2731	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
17		eqid 2731	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
18		eqid 2731	. . 3 ⊢ (-g‘𝐻) = (-g‘𝐻)
19	15, 16, 17, 18	grpsubfval 18893	. 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
20	9, 14, 19	3eqtr4g 2791	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17117  +_gcplusg 17158  inv_gcminusg 18844  -_gcsg 18845

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-0g 17342  df-minusg 18847  df-sbg 18848

This theorem is referenced by:  rlmsub  21128  matsubg  22345  tngngp2  24565  tngngp  24567  tcphsub  25146  ply1divalg2  26069  ttgsub  28855  zhmnrg  33973