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Mirrors > Home > MPE Home > Th. List > lspun0 | Structured version Visualization version GIF version |
Description: The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.) |
Ref | Expression |
---|---|
lspun0.v | β’ π = (Baseβπ) |
lspun0.o | β’ 0 = (0gβπ) |
lspun0.n | β’ π = (LSpanβπ) |
lspun0.w | β’ (π β π β LMod) |
lspun0.x | β’ (π β π β π) |
Ref | Expression |
---|---|
lspun0 | β’ (π β (πβ(π βͺ { 0 })) = (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspun0.w | . . 3 β’ (π β π β LMod) | |
2 | lspun0.x | . . 3 β’ (π β π β π) | |
3 | lspun0.v | . . . . . 6 β’ π = (Baseβπ) | |
4 | lspun0.o | . . . . . 6 β’ 0 = (0gβπ) | |
5 | 3, 4 | lmod0vcl 20735 | . . . . 5 β’ (π β LMod β 0 β π) |
6 | 1, 5 | syl 17 | . . . 4 β’ (π β 0 β π) |
7 | 6 | snssd 4807 | . . 3 β’ (π β { 0 } β π) |
8 | lspun0.n | . . . 4 β’ π = (LSpanβπ) | |
9 | 3, 8 | lspun 20832 | . . 3 β’ ((π β LMod β§ π β π β§ { 0 } β π) β (πβ(π βͺ { 0 })) = (πβ((πβπ) βͺ (πβ{ 0 })))) |
10 | 1, 2, 7, 9 | syl3anc 1368 | . 2 β’ (π β (πβ(π βͺ { 0 })) = (πβ((πβπ) βͺ (πβ{ 0 })))) |
11 | 4, 8 | lspsn0 20853 | . . . . . . 7 β’ (π β LMod β (πβ{ 0 }) = { 0 }) |
12 | 1, 11 | syl 17 | . . . . . 6 β’ (π β (πβ{ 0 }) = { 0 }) |
13 | 12 | uneq2d 4158 | . . . . 5 β’ (π β ((πβπ) βͺ (πβ{ 0 })) = ((πβπ) βͺ { 0 })) |
14 | eqid 2726 | . . . . . . . . 9 β’ (LSubSpβπ) = (LSubSpβπ) | |
15 | 3, 14, 8 | lspcl 20821 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β (πβπ) β (LSubSpβπ)) |
16 | 1, 2, 15 | syl2anc 583 | . . . . . . 7 β’ (π β (πβπ) β (LSubSpβπ)) |
17 | 4, 14 | lss0ss 20794 | . . . . . . 7 β’ ((π β LMod β§ (πβπ) β (LSubSpβπ)) β { 0 } β (πβπ)) |
18 | 1, 16, 17 | syl2anc 583 | . . . . . 6 β’ (π β { 0 } β (πβπ)) |
19 | ssequn2 4178 | . . . . . 6 β’ ({ 0 } β (πβπ) β ((πβπ) βͺ { 0 }) = (πβπ)) | |
20 | 18, 19 | sylib 217 | . . . . 5 β’ (π β ((πβπ) βͺ { 0 }) = (πβπ)) |
21 | 13, 20 | eqtrd 2766 | . . . 4 β’ (π β ((πβπ) βͺ (πβ{ 0 })) = (πβπ)) |
22 | 21 | fveq2d 6888 | . . 3 β’ (π β (πβ((πβπ) βͺ (πβ{ 0 }))) = (πβ(πβπ))) |
23 | 3, 8 | lspidm 20831 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ(πβπ)) = (πβπ)) |
24 | 1, 2, 23 | syl2anc 583 | . . 3 β’ (π β (πβ(πβπ)) = (πβπ)) |
25 | 22, 24 | eqtrd 2766 | . 2 β’ (π β (πβ((πβπ) βͺ (πβ{ 0 }))) = (πβπ)) |
26 | 10, 25 | eqtrd 2766 | 1 β’ (π β (πβ(π βͺ { 0 })) = (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βͺ cun 3941 β wss 3943 {csn 4623 βcfv 6536 Basecbs 17151 0gc0g 17392 LModclmod 20704 LSubSpclss 20776 LSpanclspn 20816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-lmod 20706 df-lss 20777 df-lsp 20817 |
This theorem is referenced by: dvh4dimN 40829 |
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