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Mirrors > Home > MPE Home > Th. List > lspsnne1 | Structured version Visualization version GIF version |
Description: Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.) |
Ref | Expression |
---|---|
lspsnne1.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnne1.o | ⊢ 0 = (0g‘𝑊) |
lspsnne1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspsnne1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspsnne1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lspsnne1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspsnne1.e | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lspsnne1 | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnne1.e | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | lspsnne1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2740 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
4 | lspsnne1.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | lspsnne1.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 20366 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
8 | lspsnne1.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
9 | 2, 3, 4 | lspsncl 20237 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
10 | 7, 8, 9 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
11 | lspsnne1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
12 | 11 | eldifad 3904 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
13 | 2, 3, 4, 7, 10, 12 | lspsnel5 20255 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}))) |
14 | 13 | notbid 318 | . . 3 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ ¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}))) |
15 | lspsnne1.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
16 | 2, 15, 4, 5, 11, 8 | lspsncmp 20376 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
17 | 16 | necon3bbid 2983 | . . 3 ⊢ (𝜑 → (¬ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))) |
18 | 14, 17 | bitrd 278 | . 2 ⊢ (𝜑 → (¬ 𝑋 ∈ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))) |
19 | 1, 18 | mpbird 256 | 1 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∖ cdif 3889 ⊆ wss 3892 {csn 4567 ‘cfv 6432 Basecbs 16910 0gc0g 17148 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 LVecclvec 20362 |
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 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
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-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 |
This theorem is referenced by: lspsnnecom 20379 lsatfixedN 37019 baerlem5amN 39726 baerlem5bmN 39727 baerlem5abmN 39728 mapdh6dN 39749 hdmaplem4 39784 hdmap1l6d 39823 hdmaprnlem3N 39860 |
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