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Mirrors > Home > MPE Home > Th. List > lsmpr | Structured version Visualization version GIF version |
Description: The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.) |
Ref | Expression |
---|---|
lsmpr.v | ⊢ 𝑉 = (Base‘𝑊) |
lsmpr.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsmpr.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsmpr.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsmpr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsmpr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lsmpr | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmpr.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lsmpr.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | 2 | snssd 4559 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lsmpr.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | 4 | snssd 4559 | . . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
6 | lsmpr.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
7 | lsmpr.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspun 19347 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
9 | 1, 3, 5, 8 | syl3anc 1496 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
10 | df-pr 4401 | . . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
11 | 10 | fveq2i 6437 | . . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))) |
13 | eqid 2826 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
14 | 6, 13, 7 | lspsncl 19337 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 1, 2, 14 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
16 | 6, 13, 7 | lspsncl 19337 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | 1, 4, 16 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
18 | lsmpr.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
19 | 13, 7, 18 | lsmsp 19446 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
20 | 1, 15, 17, 19 | syl3anc 1496 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
21 | 9, 12, 20 | 3eqtr4d 2872 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∪ cun 3797 ⊆ wss 3799 {csn 4398 {cpr 4400 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 LSSumclsm 18401 LModclmod 19220 LSubSpclss 19289 LSpanclspn 19331 |
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-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-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-cntz 18101 df-lsm 18403 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-lmod 19222 df-lss 19290 df-lsp 19332 |
This theorem is referenced by: lsppreli 19450 lsmelpr 19451 lsppr0 19452 lspprabs 19455 lspabs2 19480 lspindpi 19493 lsmsat 35084 dvh4dimlem 37519 dvh3dim3N 37525 lclkrlem2c 37585 lcfrlem20 37638 lcfrlem23 37641 mapdindp 37747 mapdindp2 37797 mapdindp4 37799 mapdh6dN 37815 lspindp5 37846 hdmap1l6d 37889 hdmaprnlem3eN 37934 |
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