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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsateln0 | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.) |
Ref | Expression |
---|---|
lsateln0.z | ⊢ 0 = (0g‘𝑊) |
lsateln0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsateln0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsateln0.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Ref | Expression |
---|---|
lsateln0 | ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsateln0.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
2 | lsateln0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2610 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lsateln0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
6 | lsateln0.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 33296 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
9 | 1, 8 | mpbid 221 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifi 3694 | . . . . . 6 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑣 ∈ (Base‘𝑊)) | |
11 | 3, 4 | lspsnid 18814 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
12 | 2, 10, 11 | syl2an 493 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
13 | eleq2 2677 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣}))) | |
14 | 12, 13 | syl5ibrcom 236 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑣 ∈ 𝑈)) |
15 | 14 | reximdva 3000 | . . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}) → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑣 ∈ 𝑈)) |
16 | 9, 15 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑣 ∈ 𝑈) |
17 | eldifsn 4260 | . . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
18 | 17 | anbi1i 727 | . . . . . 6 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈)) |
19 | anass 679 | . . . . . 6 ⊢ (((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) | |
20 | 18, 19 | bitri 263 | . . . . 5 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) |
21 | 20 | simprbi 479 | . . . 4 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈)) |
22 | 21 | ancomd 466 | . . 3 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝑈 ∧ 𝑣 ≠ 0 )) |
23 | 22 | reximi2 2993 | . 2 ⊢ (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑣 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) |
24 | 16, 23 | syl 17 | 1 ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∖ cdif 3537 {csn 4125 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LModclmod 18686 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: dvh1dim 35749 dochkr1 35785 dochkr1OLDN 35786 lcfrlem40 35889 |
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