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Mathbox for Norm Megill |
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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 473 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → 𝑊 ∈ LMod) |
5 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑁‘{𝑋}) ∈ 𝐴) | |
6 | 1, 2, 4, 5 | lsatn0 35528 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑁‘{𝑋}) ≠ { 0 }) |
7 | sneq 4445 | . . . . . . . 8 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
8 | 7 | fveq2d 6497 | . . . . . . 7 ⊢ (𝑋 = 0 → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
9 | 8 | adantl 474 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
10 | 4 | adantr 473 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → 𝑊 ∈ LMod) |
11 | lsatspn0.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 1, 11 | lspsn0 19492 | . . . . . . 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 2808 | . . . . 5 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = { 0 }) |
15 | 14 | ex 405 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑋 = 0 → (𝑁‘{𝑋}) = { 0 })) |
16 | 15 | necon3d 2982 | . . 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 473 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LMod) |
20 | isateln0.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | 20 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) |
22 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 ) | |
23 | eldifsn 4587 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
24 | 21, 22, 23 | sylanbrc 575 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
25 | 18, 11, 1, 2, 19, 24 | lsatlspsn 35522 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
26 | 17, 25 | impbida 788 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∖ cdif 3822 {csn 4435 ‘cfv 6182 Basecbs 16329 0gc0g 16559 LModclmod 19346 LSpanclspn 19455 LSAtomsclsa 35503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-plusg 16424 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-mgp 18953 df-ring 19012 df-lmod 19348 df-lss 19416 df-lsp 19456 df-lsatoms 35505 |
This theorem is referenced by: lsator0sp 35530 lcfl8b 38033 mapdpglem5N 38206 mapdpglem30a 38224 mapdpglem30b 38225 |
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