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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 3487 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
3 | fveq2 6885 | . . . . . . 7 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺)) | |
4 | lsmfval.v | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵) |
6 | 5 | pweqd 4614 | . . . . 5 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵) |
7 | fveq2 6885 | . . . . . . . . 9 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = (+g‘𝐺)) | |
8 | lsmfval.a | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | eqtr4di 2784 | . . . . . . . 8 ⊢ (𝑤 = 𝐺 → (+g‘𝑤) = + ) |
10 | 9 | oveqd 7422 | . . . . . . 7 ⊢ (𝑤 = 𝐺 → (𝑥(+g‘𝑤)𝑦) = (𝑥 + 𝑦)) |
11 | 10 | mpoeq3dv 7484 | . . . . . 6 ⊢ (𝑤 = 𝐺 → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) |
12 | 11 | rneqd 5931 | . . . . 5 ⊢ (𝑤 = 𝐺 → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) |
13 | 6, 6, 12 | mpoeq123dv 7480 | . . . 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 19556 | . . . 4 ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)))) | |
15 | 4 | fvexi 6899 | . . . . . 6 ⊢ 𝐵 ∈ V |
16 | 15 | pwex 5371 | . . . . 5 ⊢ 𝒫 𝐵 ∈ V |
17 | 16, 16 | mpoex 8065 | . . . 4 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) ∈ V |
18 | 13, 14, 17 | fvmpt 6992 | . . 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 | eqtrid 2778 | 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 1533 ∈ wcel 2098 Vcvv 3468 𝒫 cpw 4597 ran crn 5670 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 Basecbs 17153 +gcplusg 17206 LSSumclsm 19554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-lsm 19556 |
This theorem is referenced by: lsmvalx 19559 oppglsm 19562 lsmpropd 19597 rlmlsm 21061 |
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