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Mirrors > Home > MPE Home > Th. List > lssuni | Structured version Visualization version GIF version |
Description: The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.) |
Ref | Expression |
---|---|
lssss.v | ⊢ 𝑉 = (Base‘𝑊) |
lssss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssuni.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
Ref | Expression |
---|---|
lssuni | ⊢ (𝜑 → ∪ 𝑆 = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid2 3334 | . . . 4 ⊢ (𝑆 = {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} ↔ ∀𝑥 ∈ 𝑆 𝑥 ⊆ 𝑉) | |
2 | lssss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lssss.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | 2, 3 | lssss 19701 | . . . 4 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ⊆ 𝑉) |
5 | 1, 4 | mprgbir 3121 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
6 | 5 | unieqi 4813 | . 2 ⊢ ∪ 𝑆 = ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
7 | lssuni.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | 2, 3 | lss1 19703 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
9 | unimax 4836 | . . 3 ⊢ (𝑉 ∈ 𝑆 → ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} = 𝑉) | |
10 | 7, 8, 9 | 3syl 18 | . 2 ⊢ (𝜑 → ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} = 𝑉) |
11 | 6, 10 | syl5eq 2845 | 1 ⊢ (𝜑 → ∪ 𝑆 = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∪ cuni 4800 ‘cfv 6324 Basecbs 16475 LModclmod 19627 LSubSpclss 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-lmod 19629 df-lss 19697 |
This theorem is referenced by: mapdunirnN 38946 |
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