Theorem grpsubpropd 19021

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 7395	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
6	4, 1, 5	grpinvpropd 18991	. . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
7	6	fveq1d 6842	. . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏))
8	2, 3, 7	oveq123d 7388	. . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))
9	1, 1, 8	mpoeq123dv 7442	. 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 18959	. 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 18959	. 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 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  +_gcplusg 17220  inv_gcminusg 18910  -_gcsg 18911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-0g 17404  df-minusg 18913  df-sbg 18914

This theorem is referenced by:  rlmsub  21191  matsubg  22397  tngngp2  24617  tngngp  24619  tcphsub  25188  ply1divalg2  26104  ttgsub  28947  zhmnrg  34109