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Mirrors > Home > MPE Home > Th. List > lspprcl | Structured version Visualization version GIF version |
Description: The span of a pair is a subspace (frequently used special case of lspcl 20234). (Contributed by NM, 11-Apr-2015.) |
Ref | Expression |
---|---|
lspval.v | ⊢ 𝑉 = (Base‘𝑊) |
lspval.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspval.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspprcl.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspprcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspprcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lspprcl | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprcl.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lspprcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | lspprcl.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
4 | 2, 3 | prssd 4761 | . 2 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉) |
5 | lspval.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lspval.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | lspval.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 5, 6, 7 | lspcl 20234 | . 2 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆) |
9 | 1, 4, 8 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 {cpr 4569 ‘cfv 6431 Basecbs 16908 LModclmod 20119 LSubSpclss 20189 LSpanclspn 20229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-minusg 18577 df-sbg 18578 df-mgp 19717 df-ur 19734 df-ring 19781 df-lmod 20121 df-lss 20190 df-lsp 20230 |
This theorem is referenced by: lspprid1 20255 lspprvacl 20257 lsmelpr 20349 lspexch 20387 lspindpi 20390 lsppratlem4 20408 lsatfixedN 37017 dvh3dim2 39456 dvh3dim3N 39457 lclkrlem2v 39536 lcfrlem23 39573 lcfrlem25 39575 mapdindp 39679 baerlem3lem1 39715 baerlem5alem1 39716 baerlem5blem1 39717 baerlem5amN 39724 baerlem5bmN 39725 baerlem5abmN 39726 mapdh6aN 39743 mapdh6b0N 39744 mapdh6iN 39752 lspindp5 39778 mapdh8ab 39785 mapdh8ad 39787 mapdh8e 39792 mapdh9a 39797 mapdh9aOLDN 39798 hdmap1l6a 39817 hdmap1l6b0N 39818 hdmap1l6i 39826 hdmap1eulemOLDN 39831 hdmapval0 39841 hdmapval3lemN 39845 hdmap10lem 39847 hdmap11lem1 39849 hdmap11lem2 39850 hdmap14lem11 39886 |
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