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Mirrors > Home > MPE Home > Th. List > lssats2 | Structured version Visualization version GIF version |
Description: A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
lssats2.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssats2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lssats2.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssats2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Ref | Expression |
---|---|
lssats2 | ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈) | |
2 | lssats2.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | 2 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑊 ∈ LMod) |
4 | lssats2.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
5 | eqid 2798 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | lssats2.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 5, 6 | lssel 19702 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
8 | 4, 7 | sylan 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
9 | lssats2.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 5, 9 | lspsnid 19758 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (𝑁‘{𝑦})) |
11 | 3, 8, 10 | syl2anc 587 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (𝑁‘{𝑦})) |
12 | sneq 4535 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
13 | 12 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{𝑥}) = (𝑁‘{𝑦})) |
14 | 13 | eleq2d 2875 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ (𝑁‘{𝑥}) ↔ 𝑦 ∈ (𝑁‘{𝑦}))) |
15 | 14 | rspcev 3571 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑈 ∧ 𝑦 ∈ (𝑁‘{𝑦})) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
16 | 1, 11, 15 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
17 | 16 | ex 416 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑈 → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
18 | 2 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑊 ∈ LMod) |
19 | 4 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
20 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
21 | 6, 9, 18, 19, 20 | lspsnel5a 19761 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑁‘{𝑥}) ⊆ 𝑈) |
22 | 21 | sseld 3914 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
23 | 22 | rexlimdva 3243 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
24 | 17, 23 | impbid 215 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
25 | eliun 4885 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) | |
26 | 24, 25 | syl6bbr 292 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}))) |
27 | 26 | eqrdv 2796 | 1 ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {csn 4525 ∪ ciun 4881 ‘cfv 6324 Basecbs 16475 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 |
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-rep 5154 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-csb 3829 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-int 4839 df-iun 4883 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-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 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 df-lsp 19737 |
This theorem is referenced by: (None) |
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