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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 2822 | . . . 4 ⊢ (𝜑 → 𝑎 = 𝑎) | |
4 | eqidd 2822 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | 2 | oveqdr 7178 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) |
6 | 4, 1, 5 | grpinvpropd 18168 | . . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻)) |
7 | 6 | fveq1d 6666 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏)) |
8 | 2, 3, 7 | oveq123d 7171 | . . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
9 | 1, 1, 8 | mpoeq123dv 7223 | . 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 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | eqid 2821 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | eqid 2821 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
14 | 10, 11, 12, 13 | grpsubfval 18141 | . 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))) |
15 | eqid 2821 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
16 | eqid 2821 | . . 3 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
17 | eqid 2821 | . . 3 ⊢ (invg‘𝐻) = (invg‘𝐻) | |
18 | eqid 2821 | . . 3 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
19 | 15, 16, 17, 18 | grpsubfval 18141 | . 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
20 | 9, 14, 19 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 Basecbs 16477 +gcplusg 16559 invgcminusg 18098 -gcsg 18099 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-0g 16709 df-minusg 18101 df-sbg 18102 |
This theorem is referenced by: rlmsub 19964 matsubg 21035 tngngp2 23255 tngngp 23257 tcphsub 23818 ply1divalg2 24726 ttgsub 26659 zhmnrg 31203 |
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