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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 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈) | |
2 | lssats2.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑊 ∈ LMod) |
4 | lssats2.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
5 | eqid 2734 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | lssats2.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 5, 6 | lssel 20952 | . . . . . . . 8 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
8 | 4, 7 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (Base‘𝑊)) |
9 | lssats2.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 5, 9 | lspsnid 21008 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (𝑁‘{𝑦})) |
11 | 3, 8, 10 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (𝑁‘{𝑦})) |
12 | sneq 4640 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
13 | 12 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{𝑥}) = (𝑁‘{𝑦})) |
14 | 13 | eleq2d 2824 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ (𝑁‘{𝑥}) ↔ 𝑦 ∈ (𝑁‘{𝑦}))) |
15 | 14 | rspcev 3621 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑈 ∧ 𝑦 ∈ (𝑁‘{𝑦})) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
16 | 1, 11, 15 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) |
17 | 16 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑈 → ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
18 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑊 ∈ LMod) |
19 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
20 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
21 | 6, 9, 18, 19, 20 | ellspsn5 21011 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑁‘{𝑥}) ⊆ 𝑈) |
22 | 21 | sseld 3993 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
23 | 22 | rexlimdva 3152 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
24 | 17, 23 | impbid 212 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥}))) |
25 | eliun 4999 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑈 𝑦 ∈ (𝑁‘{𝑥})) | |
26 | 24, 25 | bitr4di 289 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥}))) |
27 | 26 | eqrdv 2732 | 1 ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {csn 4630 ∪ ciun 4995 ‘cfv 6562 Basecbs 17244 LModclmod 20874 LSubSpclss 20946 LSpanclspn 20986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-lmod 20876 df-lss 20947 df-lsp 20987 |
This theorem is referenced by: (None) |
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