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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 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈) | |
2 | lssats2.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | 2 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑊 ∈ LMod) |
4 | lssats2.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
5 | eqid 2821 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | lssats2.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 5, 6 | lssel 19703 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
8 | 4, 7 | sylan 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
9 | lssats2.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 5, 9 | lspsnid 19759 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (𝑁‘{𝑦})) |
11 | 3, 8, 10 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (𝑁‘{𝑦})) |
12 | sneq 4570 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
13 | 12 | fveq2d 6668 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{𝑥}) = (𝑁‘{𝑦})) |
14 | 13 | eleq2d 2898 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ (𝑁‘{𝑥}) ↔ 𝑦 ∈ (𝑁‘{𝑦}))) |
15 | 14 | rspcev 3622 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑈 ∧ 𝑦 ∈ (𝑁‘{𝑦})) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
16 | 1, 11, 15 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
17 | 16 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑈 → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
18 | 2 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑊 ∈ LMod) |
19 | 4 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
20 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
21 | 6, 9, 18, 19, 20 | lspsnel5a 19762 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑁‘{𝑥}) ⊆ 𝑈) |
22 | 21 | sseld 3965 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
23 | 22 | rexlimdva 3284 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
24 | 17, 23 | impbid 214 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
25 | eliun 4915 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) | |
26 | 24, 25 | syl6bbr 291 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}))) |
27 | 26 | eqrdv 2819 | 1 ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {csn 4560 ∪ ciun 4911 ‘cfv 6349 Basecbs 16477 LModclmod 19628 LSubSpclss 19697 LSpanclspn 19737 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-lmod 19630 df-lss 19698 df-lsp 19738 |
This theorem is referenced by: (None) |
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