![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > frlmvscaval | Structured version Visualization version GIF version |
Description: Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
frlmvscaval.y | โข ๐ = (๐ freeLMod ๐ผ) |
frlmvscaval.b | โข ๐ต = (Baseโ๐) |
frlmvscaval.k | โข ๐พ = (Baseโ๐ ) |
frlmvscaval.i | โข (๐ โ ๐ผ โ ๐) |
frlmvscaval.a | โข (๐ โ ๐ด โ ๐พ) |
frlmvscaval.x | โข (๐ โ ๐ โ ๐ต) |
frlmvscaval.j | โข (๐ โ ๐ฝ โ ๐ผ) |
frlmvscaval.v | โข โ = ( ยท๐ โ๐) |
frlmvscaval.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
frlmvscaval | โข (๐ โ ((๐ด โ ๐)โ๐ฝ) = (๐ด ยท (๐โ๐ฝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmvscaval.y | . . . 4 โข ๐ = (๐ freeLMod ๐ผ) | |
2 | frlmvscaval.b | . . . 4 โข ๐ต = (Baseโ๐) | |
3 | frlmvscaval.k | . . . 4 โข ๐พ = (Baseโ๐ ) | |
4 | frlmvscaval.i | . . . 4 โข (๐ โ ๐ผ โ ๐) | |
5 | frlmvscaval.a | . . . 4 โข (๐ โ ๐ด โ ๐พ) | |
6 | frlmvscaval.x | . . . 4 โข (๐ โ ๐ โ ๐ต) | |
7 | frlmvscaval.v | . . . 4 โข โ = ( ยท๐ โ๐) | |
8 | frlmvscaval.t | . . . 4 โข ยท = (.rโ๐ ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | frlmvscafval 21188 | . . 3 โข (๐ โ (๐ด โ ๐) = ((๐ผ ร {๐ด}) โf ยท ๐)) |
10 | 9 | fveq1d 6845 | . 2 โข (๐ โ ((๐ด โ ๐)โ๐ฝ) = (((๐ผ ร {๐ด}) โf ยท ๐)โ๐ฝ)) |
11 | fnconstg 6731 | . . . 4 โข (๐ด โ ๐พ โ (๐ผ ร {๐ด}) Fn ๐ผ) | |
12 | 5, 11 | syl 17 | . . 3 โข (๐ โ (๐ผ ร {๐ด}) Fn ๐ผ) |
13 | 1, 3, 2 | frlmbasf 21182 | . . . . 5 โข ((๐ผ โ ๐ โง ๐ โ ๐ต) โ ๐:๐ผโถ๐พ) |
14 | 4, 6, 13 | syl2anc 585 | . . . 4 โข (๐ โ ๐:๐ผโถ๐พ) |
15 | 14 | ffnd 6670 | . . 3 โข (๐ โ ๐ Fn ๐ผ) |
16 | frlmvscaval.j | . . 3 โข (๐ โ ๐ฝ โ ๐ผ) | |
17 | fnfvof 7635 | . . 3 โข ((((๐ผ ร {๐ด}) Fn ๐ผ โง ๐ Fn ๐ผ) โง (๐ผ โ ๐ โง ๐ฝ โ ๐ผ)) โ (((๐ผ ร {๐ด}) โf ยท ๐)โ๐ฝ) = (((๐ผ ร {๐ด})โ๐ฝ) ยท (๐โ๐ฝ))) | |
18 | 12, 15, 4, 16, 17 | syl22anc 838 | . 2 โข (๐ โ (((๐ผ ร {๐ด}) โf ยท ๐)โ๐ฝ) = (((๐ผ ร {๐ด})โ๐ฝ) ยท (๐โ๐ฝ))) |
19 | fvconst2g 7152 | . . . 4 โข ((๐ด โ ๐พ โง ๐ฝ โ ๐ผ) โ ((๐ผ ร {๐ด})โ๐ฝ) = ๐ด) | |
20 | 5, 16, 19 | syl2anc 585 | . . 3 โข (๐ โ ((๐ผ ร {๐ด})โ๐ฝ) = ๐ด) |
21 | 20 | oveq1d 7373 | . 2 โข (๐ โ (((๐ผ ร {๐ด})โ๐ฝ) ยท (๐โ๐ฝ)) = (๐ด ยท (๐โ๐ฝ))) |
22 | 10, 18, 21 | 3eqtrd 2777 | 1 โข (๐ โ ((๐ด โ ๐)โ๐ฝ) = (๐ด ยท (๐โ๐ฝ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 {csn 4587 ร cxp 5632 Fn wfn 6492 โถwf 6493 โcfv 6497 (class class class)co 7358 โf cof 7616 Basecbs 17088 .rcmulr 17139 ยท๐ cvsca 17142 freeLMod cfrlm 21168 |
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-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-sra 20649 df-rgmod 20650 df-dsmm 21154 df-frlm 21169 |
This theorem is referenced by: frlmvscavalb 21192 frlmvplusgscavalb 21193 frlmphl 21203 frlmssuvc2 21217 frlmup1 21220 rrxvsca 24774 frlmsnic 40771 prjspnfv01 41005 prjspner1 41007 |
Copyright terms: Public domain | W3C validator |