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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatspn0 | Structured version Visualization version GIF version |
Description: The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.) |
Ref | Expression |
---|---|
lsatspn0.v | ⊢ 𝑉 = (Base‘𝑊) |
lsatspn0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatspn0.o | ⊢ 0 = (0g‘𝑊) |
lsatspn0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
isateln0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
isateln0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lsatspn0 | ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatspn0.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
2 | lsatspn0.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
3 | isateln0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → 𝑊 ∈ LMod) |
5 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑁‘{𝑋}) ∈ 𝐴) | |
6 | 1, 2, 4, 5 | lsatn0 36699 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑁‘{𝑋}) ≠ { 0 }) |
7 | sneq 4537 | . . . . . . . 8 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
8 | 7 | fveq2d 6699 | . . . . . . 7 ⊢ (𝑋 = 0 → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
9 | 8 | adantl 485 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
10 | 4 | adantr 484 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → 𝑊 ∈ LMod) |
11 | lsatspn0.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 1, 11 | lspsn0 19999 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{ 0 }) = { 0 }) |
14 | 9, 13 | eqtrd 2771 | . . . . 5 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = { 0 }) |
15 | 14 | ex 416 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑋 = 0 → (𝑁‘{𝑋}) = { 0 })) |
16 | 15 | necon3d 2953 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → ((𝑁‘{𝑋}) ≠ { 0 } → 𝑋 ≠ 0 )) |
17 | 6, 16 | mpd 15 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → 𝑋 ≠ 0 ) |
18 | lsatspn0.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
19 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LMod) |
20 | isateln0.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | 20 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) |
22 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 ) | |
23 | eldifsn 4686 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
24 | 21, 22, 23 | sylanbrc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
25 | 18, 11, 1, 2, 19, 24 | lsatlspsn 36693 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
26 | 17, 25 | impbida 801 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 {csn 4527 ‘cfv 6358 Basecbs 16666 0gc0g 16898 LModclmod 19853 LSpanclspn 19962 LSAtomsclsa 36674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-mgp 19459 df-ring 19518 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lsatoms 36676 |
This theorem is referenced by: lsator0sp 36701 lcfl8b 39204 mapdpglem5N 39377 mapdpglem30a 39395 mapdpglem30b 39396 |
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