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Mirrors > Home > MPE Home > Th. List > lspdisj2 | Structured version Visualization version GIF version |
Description: Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
lspdisj2.v | ⊢ 𝑉 = (Base‘𝑊) |
lspdisj2.o | ⊢ 0 = (0g‘𝑊) |
lspdisj2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspdisj2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspdisj2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspdisj2.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspdisj2.q | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lspdisj2 | ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4408 | . . . . . 6 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
2 | 1 | fveq2d 6438 | . . . . 5 ⊢ (𝑋 = 0 → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
3 | lspdisj2.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19466 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | lspdisj2.o | . . . . . . 7 ⊢ 0 = (0g‘𝑊) | |
7 | lspdisj2.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspsn0 19368 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
10 | 2, 9 | sylan9eqr 2884 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = { 0 }) |
11 | 10 | ineq1d 4041 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = ({ 0 } ∩ (𝑁‘{𝑌}))) |
12 | lspdisj2.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | lspdisj2.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
14 | eqid 2826 | . . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 13, 14, 7 | lspsncl 19337 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 5, 12, 15 | syl2anc 581 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | 6, 14 | lss0ss 19306 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘{𝑌})) |
18 | 5, 16, 17 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → { 0 } ⊆ (𝑁‘{𝑌})) |
19 | df-ss 3813 | . . . . 5 ⊢ ({ 0 } ⊆ (𝑁‘{𝑌}) ↔ ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) | |
20 | 18, 19 | sylib 210 | . . . 4 ⊢ (𝜑 → ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) |
21 | 20 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) |
22 | 11, 21 | eqtrd 2862 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
23 | 3 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LVec) |
24 | 16 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
25 | lspdisj2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) |
27 | lspdisj2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
28 | 27 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
29 | 23 | adantr 474 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
30 | 12 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑌 ∈ 𝑉) |
31 | 30 | adantr 474 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑌 ∈ 𝑉) |
32 | simpr 479 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘{𝑌})) | |
33 | simplr 787 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑋 ≠ 0 ) | |
34 | 13, 6, 7, 29, 31, 32, 33 | lspsneleq 19475 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
35 | 34 | ex 403 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ (𝑁‘{𝑌}) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
36 | 35 | necon3ad 3013 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))) |
37 | 28, 36 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑌})) |
38 | 13, 6, 7, 14, 23, 24, 26, 37 | lspdisj 19485 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
39 | 22, 38 | pm2.61dane 3087 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∩ cin 3798 ⊆ wss 3799 {csn 4398 ‘cfv 6124 Basecbs 16223 0gc0g 16454 LModclmod 19220 LSubSpclss 19289 LSpanclspn 19331 LVecclvec 19462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-sbg 17782 df-mgp 18845 df-ur 18857 df-ring 18904 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-invr 19027 df-drng 19106 df-lmod 19222 df-lss 19290 df-lsp 19332 df-lvec 19463 |
This theorem is referenced by: lvecindp2 19500 hdmaprnlem9N 37933 |
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