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Mirrors > Home > MPE Home > Th. List > lsmsp2 | Structured version Visualization version GIF version |
Description: Subspace sum of spans of subsets is the span of their union. (spanuni 31367 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
Ref | Expression |
---|---|
lsmsp2.v | β’ π = (Baseβπ) |
lsmsp2.n | β’ π = (LSpanβπ) |
lsmsp2.p | β’ β = (LSSumβπ) |
Ref | Expression |
---|---|
lsmsp2 | β’ ((π β LMod β§ π β π β§ π β π) β ((πβπ) β (πβπ)) = (πβ(π βͺ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1134 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β π β LMod) | |
2 | lsmsp2.v | . . . . 5 β’ π = (Baseβπ) | |
3 | eqid 2728 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
4 | lsmsp2.n | . . . . 5 β’ π = (LSpanβπ) | |
5 | 2, 3, 4 | lspcl 20860 | . . . 4 β’ ((π β LMod β§ π β π) β (πβπ) β (LSubSpβπ)) |
6 | 5 | 3adant3 1130 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β (LSubSpβπ)) |
7 | 2, 3, 4 | lspcl 20860 | . . . 4 β’ ((π β LMod β§ π β π) β (πβπ) β (LSubSpβπ)) |
8 | 7 | 3adant2 1129 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β (LSubSpβπ)) |
9 | lsmsp2.p | . . . 4 β’ β = (LSSumβπ) | |
10 | 3, 4, 9 | lsmsp 20971 | . . 3 β’ ((π β LMod β§ (πβπ) β (LSubSpβπ) β§ (πβπ) β (LSubSpβπ)) β ((πβπ) β (πβπ)) = (πβ((πβπ) βͺ (πβπ)))) |
11 | 1, 6, 8, 10 | syl3anc 1369 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β ((πβπ) β (πβπ)) = (πβ((πβπ) βͺ (πβπ)))) |
12 | 2, 4 | lspun 20871 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβ(π βͺ π)) = (πβ((πβπ) βͺ (πβπ)))) |
13 | 11, 12 | eqtr4d 2771 | 1 β’ ((π β LMod β§ π β π β§ π β π) β ((πβπ) β (πβπ)) = (πβ(π βͺ π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βͺ cun 3945 β wss 3947 βcfv 6548 (class class class)co 7420 Basecbs 17180 LSSumclsm 19589 LModclmod 20743 LSubSpclss 20815 LSpanclspn 20855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cntz 19268 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-ur 20122 df-ring 20175 df-lmod 20745 df-lss 20816 df-lsp 20856 |
This theorem is referenced by: lsmssspx 20973 lspsntri 20982 lindsunlem 33322 dimkerim 33325 dihprrnlem1N 40897 dihprrnlem2 40898 djhlsmat 40900 dvh4dimlem 40916 lsmfgcl 42498 |
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