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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 2824 | . . . 4 ⊢ (𝜑 → 𝑎 = 𝑎) | |
4 | eqidd 2824 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | 2 | oveqdr 7186 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) |
6 | 4, 1, 5 | grpinvpropd 18176 | . . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻)) |
7 | 6 | fveq1d 6674 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏)) |
8 | 2, 3, 7 | oveq123d 7179 | . . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
9 | 1, 1, 8 | mpoeq123dv 7231 | . 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 2823 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | eqid 2823 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | eqid 2823 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | eqid 2823 | . . 3 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
14 | 10, 11, 12, 13 | grpsubfval 18149 | . 2 ⊢ (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))) |
15 | eqid 2823 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
16 | eqid 2823 | . . 3 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
17 | eqid 2823 | . . 3 ⊢ (invg‘𝐻) = (invg‘𝐻) | |
18 | eqid 2823 | . . 3 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
19 | 15, 16, 17, 18 | grpsubfval 18149 | . 2 ⊢ (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
20 | 9, 14, 19 | 3eqtr4g 2883 | 1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Basecbs 16485 +gcplusg 16567 invgcminusg 18106 -gcsg 18107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-0g 16717 df-minusg 18109 df-sbg 18110 |
This theorem is referenced by: rlmsub 19972 matsubg 21043 tngngp2 23263 tngngp 23265 tcphsub 23826 ply1divalg2 24734 ttgsub 26667 zhmnrg 31210 |
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