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Mirrors > Home > MPE Home > Th. List > lsmfval | Structured version Visualization version GIF version |
Description: The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmfval.v | ⊢ 𝐵 = (Base‘𝐺) |
lsmfval.a | ⊢ + = (+g‘𝐺) |
lsmfval.s | ⊢ ⊕ = (LSSum‘𝐺) |
Ref | Expression |
---|---|
lsmfval | ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmfval.s | . 2 ⊢ ⊕ = (LSSum‘𝐺) | |
2 | elex 3428 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
3 | fveq2 6497 | . . . . . . 7 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺)) | |
4 | lsmfval.v | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4 | syl6eqr 2827 | . . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵) |
6 | 5 | pweqd 4422 | . . . . 5 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵) |
7 | fveq2 6497 | . . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺)) | |
8 | lsmfval.a | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | syl6eqr 2827 | . . . . . . . 8 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + ) |
10 | 9 | oveqd 6992 | . . . . . . 7 ⊢ (𝑤 = 𝐺 → (𝑥(+g‘𝑤)𝑦) = (𝑥 + 𝑦)) |
11 | 10 | mpoeq3dv 7050 | . . . . . 6 ⊢ (𝑤 = 𝐺 → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) |
12 | 11 | rneqd 5649 | . . . . 5 ⊢ (𝑤 = 𝐺 → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) |
13 | 6, 6, 12 | mpoeq123dv 7046 | . . . 4 ⊢ (𝑤 = 𝐺 → (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) |
14 | df-lsm 18535 | . . . 4 ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)))) | |
15 | 4 | fvexi 6511 | . . . . . 6 ⊢ 𝐵 ∈ V |
16 | 15 | pwex 5131 | . . . . 5 ⊢ 𝒫 𝐵 ∈ V |
17 | 16, 16 | mpoex 7584 | . . . 4 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) ∈ V |
18 | 13, 14, 17 | fvmpt 6594 | . . 3 ⊢ (𝐺 ∈ V → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) |
19 | 2, 18 | syl 17 | . 2 ⊢ (𝐺 ∈ 𝑉 → (LSSum‘𝐺) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) |
20 | 1, 19 | syl5eq 2821 | 1 ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 Vcvv 3410 𝒫 cpw 4417 ran crn 5405 ‘cfv 6186 (class class class)co 6975 ∈ cmpo 6977 Basecbs 16338 +gcplusg 16420 LSSumclsm 18533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-1st 7500 df-2nd 7501 df-lsm 18535 |
This theorem is referenced by: lsmvalx 18538 oppglsm 18541 lsmpropd 18574 |
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