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Mirrors > Home > MPE Home > Th. List > grpsubpropd | Structured version Visualization version GIF version |
Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.) |
Ref | Expression |
---|---|
grpsubpropd.b | ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) |
grpsubpropd.p | ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) |
Ref | Expression |
---|---|
grpsubpropd | ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubpropd.b | . . 3 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) | |
2 | grpsubpropd.p | . . . 4 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) | |
3 | eqidd 2736 | . . . 4 ⊢ (𝜑 → 𝑎 = 𝑎) | |
4 | eqidd 2736 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | 2 | oveqdr 7459 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) |
6 | 4, 1, 5 | grpinvpropd 19046 | . . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻)) |
7 | 6 | fveq1d 6909 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏)) |
8 | 2, 3, 7 | oveq123d 7452 | . . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
9 | 1, 1, 8 | mpoeq123dv 7508 | . 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 2735 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | eqid 2735 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | eqid 2735 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | eqid 2735 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
14 | 10, 11, 12, 13 | grpsubfval 19014 | . 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))) |
15 | eqid 2735 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
16 | eqid 2735 | . . 3 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
17 | eqid 2735 | . . 3 ⊢ (invg‘𝐻) = (invg‘𝐻) | |
18 | eqid 2735 | . . 3 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
19 | 15, 16, 17, 18 | grpsubfval 19014 | . 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
20 | 9, 14, 19 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 +gcplusg 17298 invgcminusg 18965 -gcsg 18966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-0g 17488 df-minusg 18968 df-sbg 18969 |
This theorem is referenced by: rlmsub 21221 matsubg 22454 tngngp2 24689 tngngp 24691 tcphsub 25269 ply1divalg2 26193 ttgsub 28906 zhmnrg 33928 |
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