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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 4576 | . . . . . 6 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
2 | 1 | fveq2d 6772 | . . . . 5 ⊢ (𝑋 = 0 → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
3 | lspdisj2.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 20349 | . . . . . . 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 20251 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
10 | 2, 9 | sylan9eqr 2801 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = { 0 }) |
11 | 10 | ineq1d 4150 | . . 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 2739 | . . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 13, 14, 7 | lspsncl 20220 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 5, 12, 15 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | 6, 14 | lss0ss 20191 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘{𝑌})) |
18 | 5, 16, 17 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → { 0 } ⊆ (𝑁‘{𝑌})) |
19 | df-ss 3908 | . . . . 5 ⊢ ({ 0 } ⊆ (𝑁‘{𝑌}) ↔ ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) | |
20 | 18, 19 | sylib 217 | . . . 4 ⊢ (𝜑 → ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) |
21 | 20 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) |
22 | 11, 21 | eqtrd 2779 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
23 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LVec) |
24 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
25 | lspdisj2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) |
27 | lspdisj2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
28 | 27 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
29 | 23 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
30 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑌 ∈ 𝑉) |
31 | 30 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑌 ∈ 𝑉) |
32 | simpr 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘{𝑌})) | |
33 | simplr 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑋 ≠ 0 ) | |
34 | 13, 6, 7, 29, 31, 32, 33 | lspsneleq 20358 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
35 | 34 | ex 412 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ (𝑁‘{𝑌}) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
36 | 35 | necon3ad 2957 | . . . 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 20368 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
39 | 22, 38 | pm2.61dane 3033 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∩ cin 3890 ⊆ wss 3891 {csn 4566 ‘cfv 6430 Basecbs 16893 0gc0g 17131 LModclmod 20104 LSubSpclss 20174 LSpanclspn 20214 LVecclvec 20345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-sbg 18563 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 |
This theorem is referenced by: lvecindp2 20382 hdmaprnlem9N 39850 |
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