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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 18976). (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 | prssi 4353 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
5 | 2, 3, 4 | syl2anc 693 | . 2 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉) |
6 | lspval.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
7 | lspval.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
8 | lspval.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 6, 7, 8 | lspcl 18976 | . 2 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆) |
10 | 1, 5, 9 | syl2anc 693 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {cpr 4179 ‘cfv 5888 Basecbs 15857 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lsp 18972 |
This theorem is referenced by: lspprid1 18997 lspprvacl 18999 lsmelpr 19091 lspexch 19129 lspindpi 19132 lsppratlem4 19150 lsatfixedN 34296 dvh3dim2 36737 dvh3dim3N 36738 lclkrlem2v 36817 lcfrlem23 36854 lcfrlem25 36856 mapdindp 36960 baerlem3lem1 36996 baerlem5alem1 36997 baerlem5blem1 36998 baerlem5amN 37005 baerlem5bmN 37006 baerlem5abmN 37007 mapdh6aN 37024 mapdh6b0N 37025 mapdh6iN 37033 lspindp5 37059 mapdh8ab 37066 mapdh8ad 37068 mapdh8e 37073 mapdh9a 37079 mapdh9aOLDN 37080 hdmap1l6a 37099 hdmap1l6b0N 37100 hdmap1l6i 37108 hdmap1eulemOLDN 37114 hdmapval0 37125 hdmapval3lemN 37129 hdmap10lem 37131 hdmap11lem1 37133 hdmap11lem2 37134 hdmap14lem11 37170 |
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