Theorem grpsubpropd 18984

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 2731	. . . 4 ⊢ (𝜑 → 𝑎 = 𝑎)
4		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5	2	oveqdr 7418	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
6	4, 1, 5	grpinvpropd 18954	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
7	6	fveq1d 6863	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
8	2, 3, 7	oveq123d 7411	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
9	1, 1, 8	mpoeq123dv 7467	. 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 2730	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2730	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
12		eqid 2730	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
13		eqid 2730	. . 3 ⊢ (-g‘𝐺) = (-g‘𝐺)
14	10, 11, 12, 13	grpsubfval 18922	. 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))
15		eqid 2730	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
16		eqid 2730	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
17		eqid 2730	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
18		eqid 2730	. . 3 ⊢ (-g‘𝐻) = (-g‘𝐻)
19	15, 16, 17, 18	grpsubfval 18922	. 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
20	9, 14, 19	3eqtr4g 2790	1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  Basecbs 17186  +_gcplusg 17227  inv_gcminusg 18873  -_gcsg 18874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-0g 17411  df-minusg 18876  df-sbg 18877

This theorem is referenced by:  rlmsub  21110  matsubg  22326  tngngp2  24547  tngngp  24549  tcphsub  25128  ply1divalg2  26051  ttgsub  28813  zhmnrg  33962