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Mirrors > Home > MPE Home > Th. List > Mathboxes > lssatomic | Structured version Visualization version GIF version |
Description: The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 30137 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lssatomic.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssatomic.o | ⊢ 0 = (0g‘𝑊) |
lssatomic.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lssatomic.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssatomic.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lssatomic.n | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
Ref | Expression |
---|---|
lssatomic | ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssatomic.n | . . 3 ⊢ (𝜑 → 𝑈 ≠ { 0 }) | |
2 | lssatomic.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
3 | lssatomic.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
4 | lssatomic.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | 3, 4 | lssne0 19724 | . . . 4 ⊢ (𝑈 ∈ 𝑆 → (𝑈 ≠ { 0 } ↔ ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 )) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ≠ { 0 } ↔ ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 )) |
7 | 1, 6 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 ) |
8 | lssatomic.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | 8 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑊 ∈ LMod) |
10 | 2 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑈 ∈ 𝑆) |
11 | simp2 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝑈) | |
12 | eqid 2823 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
13 | 12, 4 | lssel 19711 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
14 | 10, 11, 13 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ (Base‘𝑊)) |
15 | simp3 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 ) | |
16 | eqid 2823 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | lssatomic.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
18 | 12, 16, 3, 17 | lsatlspsn2 36130 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
19 | 9, 14, 15, 18 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
20 | 4, 16, 9, 10, 11 | lspsnel5a 19770 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) |
21 | sseq1 3994 | . . . . 5 ⊢ (𝑞 = ((LSpan‘𝑊)‘{𝑥}) → (𝑞 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) | |
22 | 21 | rspcev 3625 | . . . 4 ⊢ ((((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴 ∧ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
23 | 19, 20, 22 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
24 | 23 | rexlimdv3a 3288 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑈 𝑥 ≠ 0 → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈)) |
25 | 7, 24 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 ⊆ wss 3938 {csn 4569 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LSAtomsclsa 36112 |
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 ax-un 7463 |
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 df-lsatoms 36114 |
This theorem is referenced by: lsatcvatlem 36187 dochexmidlem5 38602 |
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