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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 32377 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 20949 | . . . 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 232 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 ) |
| 8 | lssatomic.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 9 | 8 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑊 ∈ LMod) |
| 10 | 2 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑈 ∈ 𝑆) |
| 11 | simp2 1138 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝑈) | |
| 12 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 13 | 12, 4 | lssel 20935 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
| 14 | 10, 11, 13 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ (Base‘𝑊)) |
| 15 | simp3 1139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 ) | |
| 16 | eqid 2737 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
| 17 | lssatomic.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
| 18 | 12, 16, 3, 17 | lsatlspsn2 38993 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
| 19 | 9, 14, 15, 18 | syl3anc 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
| 20 | 4, 16, 9, 10, 11 | ellspsn5 20994 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) |
| 21 | sseq1 4009 | . . . . 5 ⊢ (𝑞 = ((LSpan‘𝑊)‘{𝑥}) → (𝑞 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) | |
| 22 | 21 | rspcev 3622 | . . . 4 ⊢ ((((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴 ∧ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
| 23 | 19, 20, 22 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
| 24 | 23 | rexlimdv3a 3159 | . 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3951 {csn 4626 ‘cfv 6561 Basecbs 17247 0gc0g 17484 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 LSAtomsclsa 38975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lsatoms 38977 |
| This theorem is referenced by: lsatcvatlem 39050 dochexmidlem5 41466 |
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