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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 3461 | . . . 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 20820 | . . . 4 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ⊆ 𝑉) |
5 | 1, 4 | mprgbir 3065 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
6 | 5 | unieqi 4920 | . 2 ⊢ ∪ 𝑆 = ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
7 | lssuni.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | 2, 3 | lss1 20822 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
9 | unimax 4947 | . . 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 2780 | 1 ⊢ (𝜑 → ∪ 𝑆 = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3429 ⊆ wss 3947 ∪ cuni 4908 ‘cfv 6548 Basecbs 17180 LModclmod 20743 LSubSpclss 20815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-riota 7376 df-ov 7423 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-lmod 20745 df-lss 20816 |
This theorem is referenced by: mapdunirnN 41123 |
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