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Mirrors > Home > MPE Home > Th. List > lsmelvalx | Structured version Visualization version GIF version |
Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19608. (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 |
---|---|
lsmelvalx | ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmfval.v | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | lsmfval.a | . . . 4 ⊢ + = (+g‘𝐺) | |
3 | lsmfval.s | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
4 | 1, 2, 3 | lsmvalx 19598 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧))) |
5 | 4 | eleq2d 2811 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ 𝑋 ∈ ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)))) |
6 | eqid 2725 | . . 3 ⊢ (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) = (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) | |
7 | ovex 7449 | . . 3 ⊢ (𝑦 + 𝑧) ∈ V | |
8 | 6, 7 | elrnmpo 7554 | . 2 ⊢ (𝑋 ∈ ran (𝑦 ∈ 𝑇, 𝑧 ∈ 𝑈 ↦ (𝑦 + 𝑧)) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)) |
9 | 5, 8 | bitrdi 286 | 1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ⊆ wss 3939 ran crn 5673 ‘cfv 6543 (class class class)co 7416 ∈ cmpo 7418 Basecbs 17179 +gcplusg 17232 LSSumclsm 19593 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-lsm 19595 |
This theorem is referenced by: lsmelvalix 19600 lsmless1x 19603 lsmless2x 19604 lsmelval 19608 lsmsubm 19612 lsmass 19628 lsmcomx 19815 lsmcss 21628 elgrplsmsn 33148 elringlsm 33151 lsmssass 33160 grplsm0l 33161 grplsmid 33162 mxidlprm 33232 dimkerim 33382 |
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