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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 4577 | . . . . . 6 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
2 | 1 | fveq2d 6674 | . . . . 5 ⊢ (𝑋 = 0 → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
3 | lspdisj2.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19878 | . . . . . . 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 19780 | . . . . . 6 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
10 | 2, 9 | sylan9eqr 2878 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = { 0 }) |
11 | 10 | ineq1d 4188 | . . 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 2821 | . . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 13, 14, 7 | lspsncl 19749 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 5, 12, 15 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | 6, 14 | lss0ss 19720 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → { 0 } ⊆ (𝑁‘{𝑌})) |
18 | 5, 16, 17 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → { 0 } ⊆ (𝑁‘{𝑌})) |
19 | df-ss 3952 | . . . . 5 ⊢ ({ 0 } ⊆ (𝑁‘{𝑌}) ↔ ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) | |
20 | 18, 19 | sylib 220 | . . . 4 ⊢ (𝜑 → ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) |
21 | 20 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ({ 0 } ∩ (𝑁‘{𝑌})) = { 0 }) |
22 | 11, 21 | eqtrd 2856 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
23 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LVec) |
24 | 16 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
25 | lspdisj2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
26 | 25 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) |
27 | lspdisj2.q | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
28 | 27 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
29 | 23 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
30 | 12 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑌 ∈ 𝑉) |
31 | 30 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑌 ∈ 𝑉) |
32 | simpr 487 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘{𝑌})) | |
33 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → 𝑋 ≠ 0 ) | |
34 | 13, 6, 7, 29, 31, 32, 33 | lspsneleq 19887 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ 0 ) ∧ 𝑋 ∈ (𝑁‘{𝑌})) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
35 | 34 | ex 415 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ (𝑁‘{𝑌}) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
36 | 35 | necon3ad 3029 | . . . 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 19897 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
39 | 22, 38 | pm2.61dane 3104 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ‘cfv 6355 Basecbs 16483 0gc0g 16713 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LVecclvec 19874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 |
This theorem is referenced by: lvecindp2 19911 hdmaprnlem9N 39008 |
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