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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatn0 | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 28588 analog.) (Contributed by NM, 25-Aug-2014.) |
Ref | Expression |
---|---|
lsatn0.o | ⊢ 0 = (0g‘𝑊) |
lsatn0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatn0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsatn0.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatn0 | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatn0.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
2 | lsatn0.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2610 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lsatn0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | lsatn0.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 33296 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
9 | 1, 8 | mpbid 221 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifsn 4260 | . . . . 5 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
11 | 3, 5, 4 | lspsneq0 18833 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
12 | 2, 11 | sylan 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
13 | 12 | biimpd 218 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } → 𝑣 = 0 )) |
14 | 13 | necon3d 2803 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑣 ≠ 0 → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
15 | 14 | expimpd 627 | . . . . 5 ⊢ (𝜑 → ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
16 | 10, 15 | syl5bi 231 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
17 | neeq1 2844 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ≠ { 0 } ↔ ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) | |
18 | 17 | biimprcd 239 | . . . 4 ⊢ (((LSpan‘𝑊)‘{𝑣}) ≠ { 0 } → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 })) |
19 | 16, 18 | syl6 34 | . . 3 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 }))) |
20 | 19 | rexlimdv 3012 | . 2 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 })) |
21 | 9, 20 | mpd 15 | 1 ⊢ (𝜑 → 𝑈 ≠ { 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 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ring 18372 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lsatoms 33281 |
This theorem is referenced by: lsatspn0 33305 lsatssn0 33307 lsatcmp 33308 lsatcv0 33336 dochsat 35690 dochsatshp 35758 dochshpsat 35761 dochexmidlem1 35767 |
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