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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 2622 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2622 | . . . . . 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 34278 | . . . . 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 222 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifi 3732 | . . . . . 6 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑣 ∈ (Base‘𝑊)) | |
11 | 3, 4 | lspsnid 18993 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
12 | 2, 10, 11 | syl2an 494 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
13 | eleq2 2690 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣}))) | |
14 | 12, 13 | syl5ibrcom 237 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑣 ∈ 𝑈)) |
15 | 14 | reximdva 3017 | . . 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 4317 | . . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
18 | 17 | anbi1i 731 | . . . . . 6 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈)) |
19 | anass 681 | . . . . . 6 ⊢ (((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) | |
20 | 18, 19 | bitri 264 | . . . . 5 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) |
21 | 20 | simprbi 480 | . . . 4 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈)) |
22 | 21 | ancomd 467 | . . 3 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝑈 ∧ 𝑣 ≠ 0 )) |
23 | 22 | reximi2 3010 | . 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∖ cdif 3571 {csn 4177 ‘cfv 5888 Basecbs 15857 0gc0g 16100 LModclmod 18863 LSpanclspn 18971 LSAtomsclsa 34261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lsatoms 34263 |
This theorem is referenced by: dvh1dim 36731 dochkr1 36767 dochkr1OLDN 36768 lcfrlem40 36871 |
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