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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 3383 | . . . 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 19710 | . . . 4 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ⊆ 𝑉) |
5 | 1, 4 | mprgbir 3155 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
6 | 5 | unieqi 4853 | . 2 ⊢ ∪ 𝑆 = ∪ {𝑥 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑉} |
7 | lssuni.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | 2, 3 | lss1 19712 | . . 3 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆) |
9 | unimax 4876 | . . 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 2870 | 1 ⊢ (𝜑 → ∪ 𝑆 = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 ⊆ wss 3938 ∪ cuni 4840 ‘cfv 6357 Basecbs 16485 LModclmod 19636 LSubSpclss 19705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 df-ov 7161 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-lmod 19638 df-lss 19706 |
This theorem is referenced by: mapdunirnN 38788 |
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