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Mirrors > Home > MPE Home > Th. List > frlmssuvc1 | Structured version Visualization version GIF version |
Description: A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.) |
Ref | Expression |
---|---|
frlmssuvc1.f | β’ πΉ = (π freeLMod πΌ) |
frlmssuvc1.u | β’ π = (π unitVec πΌ) |
frlmssuvc1.b | β’ π΅ = (BaseβπΉ) |
frlmssuvc1.k | β’ πΎ = (Baseβπ ) |
frlmssuvc1.t | β’ Β· = ( Β·π βπΉ) |
frlmssuvc1.z | β’ 0 = (0gβπ ) |
frlmssuvc1.c | β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} |
frlmssuvc1.r | β’ (π β π β Ring) |
frlmssuvc1.i | β’ (π β πΌ β π) |
frlmssuvc1.j | β’ (π β π½ β πΌ) |
frlmssuvc1.l | β’ (π β πΏ β π½) |
frlmssuvc1.x | β’ (π β π β πΎ) |
Ref | Expression |
---|---|
frlmssuvc1 | β’ (π β (π Β· (πβπΏ)) β πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmssuvc1.r | . . 3 β’ (π β π β Ring) | |
2 | frlmssuvc1.i | . . 3 β’ (π β πΌ β π) | |
3 | frlmssuvc1.f | . . . 4 β’ πΉ = (π freeLMod πΌ) | |
4 | 3 | frlmlmod 21171 | . . 3 β’ ((π β Ring β§ πΌ β π) β πΉ β LMod) |
5 | 1, 2, 4 | syl2anc 585 | . 2 β’ (π β πΉ β LMod) |
6 | frlmssuvc1.j | . . 3 β’ (π β π½ β πΌ) | |
7 | eqid 2733 | . . . 4 β’ (LSubSpβπΉ) = (LSubSpβπΉ) | |
8 | frlmssuvc1.b | . . . 4 β’ π΅ = (BaseβπΉ) | |
9 | frlmssuvc1.z | . . . 4 β’ 0 = (0gβπ ) | |
10 | frlmssuvc1.c | . . . 4 β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} | |
11 | 3, 7, 8, 9, 10 | frlmsslss2 21197 | . . 3 β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β πΆ β (LSubSpβπΉ)) |
12 | 1, 2, 6, 11 | syl3anc 1372 | . 2 β’ (π β πΆ β (LSubSpβπΉ)) |
13 | frlmssuvc1.x | . . 3 β’ (π β π β πΎ) | |
14 | frlmssuvc1.k | . . . 4 β’ πΎ = (Baseβπ ) | |
15 | 3 | frlmsca 21175 | . . . . . 6 β’ ((π β Ring β§ πΌ β π) β π = (ScalarβπΉ)) |
16 | 1, 2, 15 | syl2anc 585 | . . . . 5 β’ (π β π = (ScalarβπΉ)) |
17 | 16 | fveq2d 6847 | . . . 4 β’ (π β (Baseβπ ) = (Baseβ(ScalarβπΉ))) |
18 | 14, 17 | eqtrid 2785 | . . 3 β’ (π β πΎ = (Baseβ(ScalarβπΉ))) |
19 | 13, 18 | eleqtrd 2836 | . 2 β’ (π β π β (Baseβ(ScalarβπΉ))) |
20 | frlmssuvc1.u | . . . . . 6 β’ π = (π unitVec πΌ) | |
21 | 20, 3, 8 | uvcff 21213 | . . . . 5 β’ ((π β Ring β§ πΌ β π) β π:πΌβΆπ΅) |
22 | 1, 2, 21 | syl2anc 585 | . . . 4 β’ (π β π:πΌβΆπ΅) |
23 | frlmssuvc1.l | . . . . 5 β’ (π β πΏ β π½) | |
24 | 6, 23 | sseldd 3946 | . . . 4 β’ (π β πΏ β πΌ) |
25 | 22, 24 | ffvelcdmd 7037 | . . 3 β’ (π β (πβπΏ) β π΅) |
26 | 3, 14, 8 | frlmbasf 21182 | . . . . 5 β’ ((πΌ β π β§ (πβπΏ) β π΅) β (πβπΏ):πΌβΆπΎ) |
27 | 2, 25, 26 | syl2anc 585 | . . . 4 β’ (π β (πβπΏ):πΌβΆπΎ) |
28 | 1 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β (πΌ β π½)) β π β Ring) |
29 | 2 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β (πΌ β π½)) β πΌ β π) |
30 | 24 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β (πΌ β π½)) β πΏ β πΌ) |
31 | eldifi 4087 | . . . . . 6 β’ (π₯ β (πΌ β π½) β π₯ β πΌ) | |
32 | 31 | adantl 483 | . . . . 5 β’ ((π β§ π₯ β (πΌ β π½)) β π₯ β πΌ) |
33 | disjdif 4432 | . . . . . 6 β’ (π½ β© (πΌ β π½)) = β | |
34 | simpr 486 | . . . . . 6 β’ ((π β§ π₯ β (πΌ β π½)) β π₯ β (πΌ β π½)) | |
35 | disjne 4415 | . . . . . 6 β’ (((π½ β© (πΌ β π½)) = β β§ πΏ β π½ β§ π₯ β (πΌ β π½)) β πΏ β π₯) | |
36 | 33, 23, 34, 35 | mp3an2ani 1469 | . . . . 5 β’ ((π β§ π₯ β (πΌ β π½)) β πΏ β π₯) |
37 | 20, 28, 29, 30, 32, 36, 9 | uvcvv0 21212 | . . . 4 β’ ((π β§ π₯ β (πΌ β π½)) β ((πβπΏ)βπ₯) = 0 ) |
38 | 27, 37 | suppss 8126 | . . 3 β’ (π β ((πβπΏ) supp 0 ) β π½) |
39 | oveq1 7365 | . . . . 5 β’ (π₯ = (πβπΏ) β (π₯ supp 0 ) = ((πβπΏ) supp 0 )) | |
40 | 39 | sseq1d 3976 | . . . 4 β’ (π₯ = (πβπΏ) β ((π₯ supp 0 ) β π½ β ((πβπΏ) supp 0 ) β π½)) |
41 | 40, 10 | elrab2 3649 | . . 3 β’ ((πβπΏ) β πΆ β ((πβπΏ) β π΅ β§ ((πβπΏ) supp 0 ) β π½)) |
42 | 25, 38, 41 | sylanbrc 584 | . 2 β’ (π β (πβπΏ) β πΆ) |
43 | eqid 2733 | . . 3 β’ (ScalarβπΉ) = (ScalarβπΉ) | |
44 | frlmssuvc1.t | . . 3 β’ Β· = ( Β·π βπΉ) | |
45 | eqid 2733 | . . 3 β’ (Baseβ(ScalarβπΉ)) = (Baseβ(ScalarβπΉ)) | |
46 | 43, 44, 45, 7 | lssvscl 20431 | . 2 β’ (((πΉ β LMod β§ πΆ β (LSubSpβπΉ)) β§ (π β (Baseβ(ScalarβπΉ)) β§ (πβπΏ) β πΆ)) β (π Β· (πβπΏ)) β πΆ) |
47 | 5, 12, 19, 42, 46 | syl22anc 838 | 1 β’ (π β (π Β· (πβπΏ)) β πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 {crab 3406 β cdif 3908 β© cin 3910 β wss 3911 β c0 4283 βΆwf 6493 βcfv 6497 (class class class)co 7358 supp csupp 8093 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 Ringcrg 19969 LModclmod 20336 LSubSpclss 20407 freeLMod cfrlm 21168 unitVec cuvc 21204 |
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-tp 4592 df-op 4594 df-uni 4867 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-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 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-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-prds 17334 df-pws 17336 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-ghm 19011 df-mgp 19902 df-ur 19919 df-ring 19971 df-subrg 20234 df-lmod 20338 df-lss 20408 df-lmhm 20498 df-sra 20649 df-rgmod 20650 df-dsmm 21154 df-frlm 21169 df-uvc 21205 |
This theorem is referenced by: (None) |
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