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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 2823 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | lssats2.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 5, 6 | lssel 19711 | . . . . . . . 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 19767 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (𝑁‘{𝑦})) |
11 | 3, 8, 10 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ (𝑁‘{𝑦})) |
12 | sneq 4579 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
13 | 12 | fveq2d 6676 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑁‘{𝑥}) = (𝑁‘{𝑦})) |
14 | 13 | eleq2d 2900 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ (𝑁‘{𝑥}) ↔ 𝑦 ∈ (𝑁‘{𝑦}))) |
15 | 14 | rspcev 3625 | . . . . . 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 19770 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑁‘{𝑥}) ⊆ 𝑈) |
22 | 21 | sseld 3968 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑦 ∈ (𝑁‘{𝑥}) → 𝑦 ∈ 𝑈)) |
23 | 22 | rexlimdva 3286 | . . . 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 4925 | . . 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 2821 | 1 ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 {csn 4569 ∪ ciun 4921 ‘cfv 6357 Basecbs 16485 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 |
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-rep 5192 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-csb 3886 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-int 4879 df-iun 4923 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-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 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 df-lsp 19746 |
This theorem is referenced by: (None) |
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