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| Mirrors > Home > MPE Home > Th. List > lsmvalx | Structured version Visualization version GIF version | ||
| Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 19666. (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 |
|---|---|
| lsmvalx | ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lsmfval.v | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | lsmfval.a | . . . . 5 ⊢ + = (+g‘𝐺) | |
| 3 | lsmfval.s | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
| 4 | 1, 2, 3 | lsmfval 19656 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))) |
| 5 | 4 | oveqd 7448 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑇 ⊕ 𝑈) = (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))𝑈)) |
| 6 | 1 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 7 | 6 | elpw2 5334 | . . . 4 ⊢ (𝑇 ∈ 𝒫 𝐵 ↔ 𝑇 ⊆ 𝐵) |
| 8 | 6 | elpw2 5334 | . . . 4 ⊢ (𝑈 ∈ 𝒫 𝐵 ↔ 𝑈 ⊆ 𝐵) |
| 9 | mpoexga 8102 | . . . . . 6 ⊢ ((𝑇 ∈ 𝒫 𝐵 ∧ 𝑈 ∈ 𝒫 𝐵) → (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)) ∈ V) | |
| 10 | rnexg 7924 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)) ∈ V → ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)) ∈ V) | |
| 11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝑇 ∈ 𝒫 𝐵 ∧ 𝑈 ∈ 𝒫 𝐵) → ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)) ∈ V) |
| 12 | mpoeq12 7506 | . . . . . . 7 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) | |
| 13 | 12 | rneqd 5949 | . . . . . 6 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| 14 | eqid 2737 | . . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))) | |
| 15 | 13, 14 | ovmpoga 7587 | . . . . 5 ⊢ ((𝑇 ∈ 𝒫 𝐵 ∧ 𝑈 ∈ 𝒫 𝐵 ∧ ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)) ∈ V) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| 16 | 11, 15 | mpd3an3 1464 | . . . 4 ⊢ ((𝑇 ∈ 𝒫 𝐵 ∧ 𝑈 ∈ 𝒫 𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| 17 | 7, 8, 16 | syl2anbr 599 | . . 3 ⊢ ((𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇(𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦)))𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| 18 | 5, 17 | sylan9eq 2797 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| 19 | 18 | 3impb 1115 | 1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +gcplusg 17297 LSSumclsm 19652 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-lsm 19654 |
| This theorem is referenced by: lsmelvalx 19658 lsmssv 19661 lsmval 19666 smndlsmidm 19674 subglsm 19691 lsmssass 33430 |
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