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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 32291 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 20928 | . . . 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 231 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 ) |
8 | lssatomic.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | 8 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑊 ∈ LMod) |
10 | 2 | 3ad2ant1 1130 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑈 ∈ 𝑆) |
11 | simp2 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝑈) | |
12 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
13 | 12, 4 | lssel 20914 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
14 | 10, 11, 13 | syl2anc 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ (Base‘𝑊)) |
15 | simp3 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 ) | |
16 | eqid 2726 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | lssatomic.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
18 | 12, 16, 3, 17 | lsatlspsn2 38690 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
19 | 9, 14, 15, 18 | syl3anc 1368 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
20 | 4, 16, 9, 10, 11 | ellspsn5 20973 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) |
21 | sseq1 4005 | . . . . 5 ⊢ (𝑞 = ((LSpan‘𝑊)‘{𝑥}) → (𝑞 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) | |
22 | 21 | rspcev 3608 | . . . 4 ⊢ ((((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴 ∧ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
23 | 19, 20, 22 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
24 | 23 | rexlimdv3a 3149 | . 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 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ⊆ wss 3947 {csn 4633 ‘cfv 6554 Basecbs 17213 0gc0g 17454 LModclmod 20836 LSubSpclss 20908 LSpanclspn 20948 LSAtomsclsa 38672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lsatoms 38674 |
This theorem is referenced by: lsatcvatlem 38747 dochexmidlem5 41163 |
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