Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 28601 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 18772 | . . . 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 221 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 𝑥 ≠ 0 ) |
8 | lssatomic.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | 8 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑊 ∈ LMod) |
10 | 2 | 3ad2ant1 1075 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑈 ∈ 𝑆) |
11 | simp2 1055 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝑈) | |
12 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
13 | 12, 4 | lssel 18759 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑊)) |
14 | 10, 11, 13 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ (Base‘𝑊)) |
15 | simp3 1056 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 ) | |
16 | eqid 2610 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | lssatomic.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
18 | 12, 16, 3, 17 | lsatlspsn2 33297 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
19 | 9, 14, 15, 18 | syl3anc 1318 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴) |
20 | 4, 16, 9, 10, 11 | lspsnel5a 18817 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) |
21 | sseq1 3589 | . . . . 5 ⊢ (𝑞 = ((LSpan‘𝑊)‘{𝑥}) → (𝑞 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) | |
22 | 21 | rspcev 3282 | . . . 4 ⊢ ((((LSpan‘𝑊)‘{𝑥}) ∈ 𝐴 ∧ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
23 | 19, 20, 22 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈 ∧ 𝑥 ≠ 0 ) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈) |
24 | 23 | rexlimdv3a 3015 | . 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 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ⊆ wss 3540 {csn 4125 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LSAtomsclsa 33279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lsatoms 33281 |
This theorem is referenced by: lsatcvatlem 33354 dochexmidlem5 35771 |
Copyright terms: Public domain | W3C validator |