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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 29742 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 19269 | . . . 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 224 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 ) |
8 | lssatomic.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | 8 | 3ad2ant1 1164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑊 ∈ LMod) |
10 | 2 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑈 ∈ 𝑆) |
11 | simp2 1168 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝑈) | |
12 | eqid 2799 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
13 | 12, 4 | lssel 19256 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
14 | 10, 11, 13 | syl2anc 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ (Base‘𝑊)) |
15 | simp3 1169 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 ) | |
16 | eqid 2799 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | lssatomic.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
18 | 12, 16, 3, 17 | lsatlspsn2 35013 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
19 | 9, 14, 15, 18 | syl3anc 1491 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
20 | 4, 16, 9, 10, 11 | lspsnel5a 19317 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) |
21 | sseq1 3822 | . . . . 5 ⊢ (𝑞 = ((LSpan‘𝑊)‘{𝑥}) → (𝑞 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) | |
22 | 21 | rspcev 3497 | . . . 4 ⊢ ((((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴 ∧ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
23 | 19, 20, 22 | syl2anc 580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
24 | 23 | rexlimdv3a 3214 | . 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 198 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∃wrex 3090 ⊆ wss 3769 {csn 4368 ‘cfv 6101 Basecbs 16184 0gc0g 16415 LModclmod 19181 LSubSpclss 19250 LSpanclspn 19292 LSAtomsclsa 34995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lsatoms 34997 |
This theorem is referenced by: lsatcvatlem 35070 dochexmidlem5 37485 |
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