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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatspn0 | Structured version Visualization version GIF version |
Description: The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.) |
Ref | Expression |
---|---|
lsatspn0.v | ⊢ 𝑉 = (Base‘𝑊) |
lsatspn0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatspn0.o | ⊢ 0 = (0g‘𝑊) |
lsatspn0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
isateln0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
isateln0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lsatspn0 | ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatspn0.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
2 | lsatspn0.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
3 | isateln0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → 𝑊 ∈ LMod) |
5 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑁‘{𝑋}) ∈ 𝐴) | |
6 | 1, 2, 4, 5 | lsatn0 36137 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑁‘{𝑋}) ≠ { 0 }) |
7 | sneq 4579 | . . . . . . . 8 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
8 | 7 | fveq2d 6676 | . . . . . . 7 ⊢ (𝑋 = 0 → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
9 | 8 | adantl 484 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{ 0 })) |
10 | 4 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → 𝑊 ∈ LMod) |
11 | lsatspn0.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | 1, 11 | lspsn0 19782 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
13 | 10, 12 | syl 17 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{ 0 }) = { 0 }) |
14 | 9, 13 | eqtrd 2858 | . . . . 5 ⊢ (((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝑁‘{𝑋}) = { 0 }) |
15 | 14 | ex 415 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → (𝑋 = 0 → (𝑁‘{𝑋}) = { 0 })) |
16 | 15 | necon3d 3039 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → ((𝑁‘{𝑋}) ≠ { 0 } → 𝑋 ≠ 0 )) |
17 | 6, 16 | mpd 15 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑋}) ∈ 𝐴) → 𝑋 ≠ 0 ) |
18 | lsatspn0.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
19 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑊 ∈ LMod) |
20 | isateln0.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
21 | 20 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉) |
22 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 ) | |
23 | eldifsn 4721 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
24 | 21, 22, 23 | sylanbrc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
25 | 18, 11, 1, 2, 19, 24 | lsatlspsn 36131 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴) |
26 | 17, 25 | impbida 799 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ‘cfv 6357 Basecbs 16485 0gc0g 16715 LModclmod 19636 LSpanclspn 19745 LSAtomsclsa 36112 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-mgp 19242 df-ring 19301 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lsatoms 36114 |
This theorem is referenced by: lsator0sp 36139 lcfl8b 38642 mapdpglem5N 38815 mapdpglem30a 38833 mapdpglem30b 38834 |
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