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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 3458 | . . . 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 20781 | . . . 4 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ⊆ 𝑉) |
5 | 1, 4 | mprgbir 3062 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
6 | 5 | unieqi 4914 | . 2 ⊢ ∪ 𝑆 = ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
7 | lssuni.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | 2, 3 | lss1 20783 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
9 | unimax 4941 | . . 3 ⊢ (𝑉 ∈ 𝑆 → ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} = 𝑉) | |
10 | 7, 8, 9 | 3syl 18 | . 2 ⊢ (𝜑 → ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} = 𝑉) |
11 | 6, 10 | eqtrid 2778 | 1 ⊢ (𝜑 → ∪ 𝑆 = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3426 ⊆ wss 3943 ∪ cuni 4902 ‘cfv 6536 Basecbs 17151 LModclmod 20704 LSubSpclss 20776 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
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-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 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-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-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-lmod 20706 df-lss 20777 |
This theorem is referenced by: mapdunirnN 41032 |
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