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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 3468 | . . . 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 20952 | . . . 4 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ⊆ 𝑉) |
5 | 1, 4 | mprgbir 3066 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
6 | 5 | unieqi 4924 | . 2 ⊢ ∪ 𝑆 = ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
7 | lssuni.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | 2, 3 | lss1 20954 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
9 | unimax 4949 | . . 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 2787 | 1 ⊢ (𝜑 → ∪ 𝑆 = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 ∪ cuni 4912 ‘cfv 6563 Basecbs 17245 LModclmod 20875 LSubSpclss 20947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-lmod 20877 df-lss 20948 |
This theorem is referenced by: mapdunirnN 41633 |
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