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Mirrors > Home > MPE Home > Th. List > lsppr | Structured version Visualization version GIF version |
Description: Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.) |
Ref | Expression |
---|---|
lsppr.v | ⊢ 𝑉 = (Base‘𝑊) |
lsppr.a | ⊢ + = (+g‘𝑊) |
lsppr.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lsppr.k | ⊢ 𝐾 = (Base‘𝐹) |
lsppr.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lsppr.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsppr.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsppr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsppr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lsppr | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4400 | . . 3 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
2 | 1 | fveq2i 6436 | . 2 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
3 | lsppr.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lsppr.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
5 | 4 | snssd 4558 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
6 | lsppr.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | 6 | snssd 4558 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
8 | lsppr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | lsppr.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 8, 9 | lspun 19346 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
11 | 3, 5, 7, 10 | syl3anc 1496 | . . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
12 | eqid 2825 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | 8, 12, 9 | lspsncl 19336 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
14 | 3, 4, 13 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 8, 12, 9 | lspsncl 19336 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 3, 6, 15 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | eqid 2825 | . . . . 5 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
18 | 12, 9, 17 | lsmsp 19445 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
19 | 3, 14, 16, 18 | syl3anc 1496 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
20 | lsppr.a | . . . . 5 ⊢ + = (+g‘𝑊) | |
21 | lsppr.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
22 | lsppr.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
23 | lsppr.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
24 | 8, 20, 21, 22, 23, 17, 9, 3, 4, 6 | lsmspsn 19443 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
25 | 24 | abbi2dv 2947 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
26 | 11, 19, 25 | 3eqtr2d 2867 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
27 | 2, 26 | syl5eq 2873 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {cab 2811 ∃wrex 3118 ∪ cun 3796 ⊆ wss 3798 {csn 4397 {cpr 4399 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 Scalarcsca 16308 ·𝑠 cvsca 16309 LSSumclsm 18400 LModclmod 19219 LSubSpclss 19288 LSpanclspn 19330 |
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 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
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 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-lmod 19221 df-lss 19289 df-lsp 19331 |
This theorem is referenced by: lspprel 19453 |
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