![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > lspun0 | 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 13445 | . . . . 5 β’ (π β LMod β 0 β π) |
6 | 1, 5 | syl 14 | . . . 4 β’ (π β 0 β π) |
7 | 6 | snssd 3739 | . . 3 β’ (π β { 0 } β π) |
8 | lspun0.n | . . . 4 β’ π = (LSpanβπ) | |
9 | 3, 8 | lspun 13527 | . . 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 1238 | . 2 β’ (π β (πβ(π βͺ { 0 })) = (πβ((πβπ) βͺ (πβ{ 0 })))) |
11 | 4, 8 | lspsn0 13547 | . . . . . . 7 β’ (π β LMod β (πβ{ 0 }) = { 0 }) |
12 | 1, 11 | syl 14 | . . . . . 6 β’ (π β (πβ{ 0 }) = { 0 }) |
13 | 12 | uneq2d 3291 | . . . . 5 β’ (π β ((πβπ) βͺ (πβ{ 0 })) = ((πβπ) βͺ { 0 })) |
14 | eqid 2177 | . . . . . . . . 9 β’ (LSubSpβπ) = (LSubSpβπ) | |
15 | 3, 14, 8 | lspcl 13516 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β (πβπ) β (LSubSpβπ)) |
16 | 1, 2, 15 | syl2anc 411 | . . . . . . 7 β’ (π β (πβπ) β (LSubSpβπ)) |
17 | 4, 14 | lss0ss 13496 | . . . . . . 7 β’ ((π β LMod β§ (πβπ) β (LSubSpβπ)) β { 0 } β (πβπ)) |
18 | 1, 16, 17 | syl2anc 411 | . . . . . 6 β’ (π β { 0 } β (πβπ)) |
19 | ssequn2 3310 | . . . . . 6 β’ ({ 0 } β (πβπ) β ((πβπ) βͺ { 0 }) = (πβπ)) | |
20 | 18, 19 | sylib 122 | . . . . 5 β’ (π β ((πβπ) βͺ { 0 }) = (πβπ)) |
21 | 13, 20 | eqtrd 2210 | . . . 4 β’ (π β ((πβπ) βͺ (πβ{ 0 })) = (πβπ)) |
22 | 21 | fveq2d 5521 | . . 3 β’ (π β (πβ((πβπ) βͺ (πβ{ 0 }))) = (πβ(πβπ))) |
23 | 3, 8 | lspidm 13526 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ(πβπ)) = (πβπ)) |
24 | 1, 2, 23 | syl2anc 411 | . . 3 β’ (π β (πβ(πβπ)) = (πβπ)) |
25 | 22, 24 | eqtrd 2210 | . 2 β’ (π β (πβ((πβπ) βͺ (πβ{ 0 }))) = (πβπ)) |
26 | 10, 25 | eqtrd 2210 | 1 β’ (π β (πβ(π βͺ { 0 })) = (πβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 βͺ cun 3129 β wss 3131 {csn 3594 βcfv 5218 Basecbs 12465 0gc0g 12711 LModclmod 13415 LSubSpclss 13480 LSpanclspn 13511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-6 8985 df-ndx 12468 df-slot 12469 df-base 12471 df-sets 12472 df-plusg 12552 df-mulr 12553 df-sca 12555 df-vsca 12556 df-0g 12713 df-mgm 12782 df-sgrp 12815 df-mnd 12826 df-grp 12888 df-minusg 12889 df-sbg 12890 df-mgp 13147 df-ur 13181 df-ring 13219 df-lmod 13417 df-lssm 13481 df-lsp 13512 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |