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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 20366 | . . . . 5 β’ (π β LMod β 0 β π) |
6 | 1, 5 | syl 17 | . . . 4 β’ (π β 0 β π) |
7 | 6 | snssd 4770 | . . 3 β’ (π β { 0 } β π) |
8 | lspun0.n | . . . 4 β’ π = (LSpanβπ) | |
9 | 3, 8 | lspun 20463 | . . 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 1372 | . 2 β’ (π β (πβ(π βͺ { 0 })) = (πβ((πβπ) βͺ (πβ{ 0 })))) |
11 | 4, 8 | lspsn0 20484 | . . . . . . 7 β’ (π β LMod β (πβ{ 0 }) = { 0 }) |
12 | 1, 11 | syl 17 | . . . . . 6 β’ (π β (πβ{ 0 }) = { 0 }) |
13 | 12 | uneq2d 4124 | . . . . 5 β’ (π β ((πβπ) βͺ (πβ{ 0 })) = ((πβπ) βͺ { 0 })) |
14 | eqid 2733 | . . . . . . . . 9 β’ (LSubSpβπ) = (LSubSpβπ) | |
15 | 3, 14, 8 | lspcl 20452 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β (πβπ) β (LSubSpβπ)) |
16 | 1, 2, 15 | syl2anc 585 | . . . . . . 7 β’ (π β (πβπ) β (LSubSpβπ)) |
17 | 4, 14 | lss0ss 20424 | . . . . . . 7 β’ ((π β LMod β§ (πβπ) β (LSubSpβπ)) β { 0 } β (πβπ)) |
18 | 1, 16, 17 | syl2anc 585 | . . . . . 6 β’ (π β { 0 } β (πβπ)) |
19 | ssequn2 4144 | . . . . . 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 2773 | . . . 4 β’ (π β ((πβπ) βͺ (πβ{ 0 })) = (πβπ)) |
22 | 21 | fveq2d 6847 | . . 3 β’ (π β (πβ((πβπ) βͺ (πβ{ 0 }))) = (πβ(πβπ))) |
23 | 3, 8 | lspidm 20462 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ(πβπ)) = (πβπ)) |
24 | 1, 2, 23 | syl2anc 585 | . . 3 β’ (π β (πβ(πβπ)) = (πβπ)) |
25 | 22, 24 | eqtrd 2773 | . 2 β’ (π β (πβ((πβπ) βͺ (πβ{ 0 }))) = (πβπ)) |
26 | 10, 25 | eqtrd 2773 | 1 β’ (π β (πβ(π βͺ { 0 })) = (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cun 3909 β wss 3911 {csn 4587 βcfv 6497 Basecbs 17088 0gc0g 17326 LModclmod 20336 LSubSpclss 20407 LSpanclspn 20447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: dvh4dimN 39956 |
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