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Mirrors > Home > MPE Home > Th. List > pwsvscaval | Structured version Visualization version GIF version |
Description: Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsvscaval.y | β’ π = (π βs πΌ) |
pwsvscaval.b | β’ π΅ = (Baseβπ) |
pwsvscaval.s | β’ Β· = ( Β·π βπ ) |
pwsvscaval.t | β’ β = ( Β·π βπ) |
pwsvscaval.f | β’ πΉ = (Scalarβπ ) |
pwsvscaval.k | β’ πΎ = (BaseβπΉ) |
pwsvscaval.r | β’ (π β π β π) |
pwsvscaval.i | β’ (π β πΌ β π) |
pwsvscaval.a | β’ (π β π΄ β πΎ) |
pwsvscaval.x | β’ (π β π β π΅) |
pwsvscaval.j | β’ (π β π½ β πΌ) |
Ref | Expression |
---|---|
pwsvscaval | β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsvscaval.y | . . . 4 β’ π = (π βs πΌ) | |
2 | pwsvscaval.b | . . . 4 β’ π΅ = (Baseβπ) | |
3 | pwsvscaval.s | . . . 4 β’ Β· = ( Β·π βπ ) | |
4 | pwsvscaval.t | . . . 4 β’ β = ( Β·π βπ) | |
5 | pwsvscaval.f | . . . 4 β’ πΉ = (Scalarβπ ) | |
6 | pwsvscaval.k | . . . 4 β’ πΎ = (BaseβπΉ) | |
7 | pwsvscaval.r | . . . 4 β’ (π β π β π) | |
8 | pwsvscaval.i | . . . 4 β’ (π β πΌ β π) | |
9 | pwsvscaval.a | . . . 4 β’ (π β π΄ β πΎ) | |
10 | pwsvscaval.x | . . . 4 β’ (π β π β π΅) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | pwsvscafval 17470 | . . 3 β’ (π β (π΄ β π) = ((πΌ Γ {π΄}) βf Β· π)) |
12 | 11 | fveq1d 6892 | . 2 β’ (π β ((π΄ β π)βπ½) = (((πΌ Γ {π΄}) βf Β· π)βπ½)) |
13 | pwsvscaval.j | . . 3 β’ (π β π½ β πΌ) | |
14 | eqid 2725 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
15 | 1, 14, 2, 7, 8, 10 | pwselbas 17465 | . . . . 5 β’ (π β π:πΌβΆ(Baseβπ )) |
16 | 15 | ffnd 6718 | . . . 4 β’ (π β π Fn πΌ) |
17 | eqidd 2726 | . . . 4 β’ ((π β§ π½ β πΌ) β (πβπ½) = (πβπ½)) | |
18 | 8, 9, 16, 17 | ofc1 7706 | . . 3 β’ ((π β§ π½ β πΌ) β (((πΌ Γ {π΄}) βf Β· π)βπ½) = (π΄ Β· (πβπ½))) |
19 | 13, 18 | mpdan 685 | . 2 β’ (π β (((πΌ Γ {π΄}) βf Β· π)βπ½) = (π΄ Β· (πβπ½))) |
20 | 12, 19 | eqtrd 2765 | 1 β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {csn 4625 Γ cxp 5671 βcfv 6543 (class class class)co 7413 βf cof 7677 Basecbs 17174 Scalarcsca 17230 Β·π cvsca 17231 βs cpws 17422 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-prds 17423 df-pws 17425 |
This theorem is referenced by: (None) |
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