Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatdim | Structured version Visualization version GIF version |
Description: A line, spanned by a nonzero singleton, has dimension 1. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
---|---|
lbslsat.v | ⊢ 𝑉 = (Base‘𝑊) |
lbslsat.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lbslsat.z | ⊢ 0 = (0g‘𝑊) |
lbslsat.y | ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋})) |
Ref | Expression |
---|---|
lsatdim | ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (dim‘𝑌) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LVec) | |
2 | lveclmod 20364 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LMod) |
4 | simp2 1136 | . . . . . 6 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) | |
5 | 4 | snssd 4748 | . . . . 5 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → {𝑋} ⊆ 𝑉) |
6 | lbslsat.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
7 | eqid 2740 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | lbslsat.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 6, 7, 8 | lspcl 20234 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
10 | 3, 5, 9 | syl2anc 584 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
11 | lbslsat.y | . . . . 5 ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋})) | |
12 | 11, 7 | lsslvec 20365 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) → 𝑌 ∈ LVec) |
13 | 1, 10, 12 | syl2anc 584 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → 𝑌 ∈ LVec) |
14 | lbslsat.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
15 | 6, 8, 14, 11 | lbslsat 31693 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → {𝑋} ∈ (LBasis‘𝑌)) |
16 | eqid 2740 | . . . 4 ⊢ (LBasis‘𝑌) = (LBasis‘𝑌) | |
17 | 16 | dimval 31680 | . . 3 ⊢ ((𝑌 ∈ LVec ∧ {𝑋} ∈ (LBasis‘𝑌)) → (dim‘𝑌) = (♯‘{𝑋})) |
18 | 13, 15, 17 | syl2anc 584 | . 2 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (dim‘𝑌) = (♯‘{𝑋})) |
19 | hashsng 14080 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (♯‘{𝑋}) = 1) | |
20 | 4, 19 | syl 17 | . 2 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (♯‘{𝑋}) = 1) |
21 | 18, 20 | eqtrd 2780 | 1 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (dim‘𝑌) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ⊆ wss 3892 {csn 4567 ‘cfv 6431 (class class class)co 7269 1c1 10871 ♯chash 14040 Basecbs 16908 ↾s cress 16937 0gc0g 17146 LModclmod 20119 LSubSpclss 20189 LSpanclspn 20229 LBasisclbs 20332 LVecclvec 20360 dimcldim 31678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-reg 9327 ax-inf2 9375 ax-ac2 10218 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-tpos 8031 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-oi 9245 df-r1 9521 df-rank 9522 df-card 9696 df-acn 9699 df-ac 9871 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-xnn0 12304 df-z 12318 df-dec 12435 df-uz 12580 df-fz 13237 df-hash 14041 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-tset 16977 df-ple 16978 df-ocomp 16979 df-0g 17148 df-mre 17291 df-mrc 17292 df-mri 17293 df-acs 17294 df-proset 18009 df-drs 18010 df-poset 18027 df-ipo 18242 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-grp 18576 df-minusg 18577 df-sbg 18578 df-subg 18748 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-oppr 19858 df-dvdsr 19879 df-unit 19880 df-invr 19910 df-drng 19989 df-lmod 20121 df-lss 20190 df-lsp 20230 df-lbs 20333 df-lvec 20361 df-dim 31679 |
This theorem is referenced by: drngdimgt0 31695 |
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