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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 2732 | . . . 4 ⊢ (𝜑 → 𝑎 = 𝑎) | |
4 | eqidd 2732 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | 2 | oveqdr 7421 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) |
6 | 4, 1, 5 | grpinvpropd 18872 | . . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻)) |
7 | 6 | fveq1d 6880 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏)) |
8 | 2, 3, 7 | oveq123d 7414 | . . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
9 | 1, 1, 8 | mpoeq123dv 7468 | . 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 18843 | . 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 18843 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 Basecbs 17126 +gcplusg 17179 invgcminusg 18795 -gcsg 18796 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-0g 17369 df-minusg 18798 df-sbg 18799 |
This theorem is referenced by: rlmsub 20768 matsubg 21863 tngngp2 24098 tngngp 24100 tcphsub 24667 ply1divalg2 25585 ttgsub 27999 zhmnrg 32776 |
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