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Mirrors > Home > MPE Home > Th. List > rrxvsca | Structured version Visualization version GIF version |
Description: The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.) |
Ref | Expression |
---|---|
rrxval.r | β’ π» = (β^βπΌ) |
rrxbase.b | β’ π΅ = (Baseβπ») |
rrxvsca.r | β’ β = ( Β·π βπ») |
rrxvsca.i | β’ (π β πΌ β π) |
rrxvsca.j | β’ (π β π½ β πΌ) |
rrxvsca.a | β’ (π β π΄ β β) |
rrxvsca.x | β’ (π β π β (Baseβπ»)) |
Ref | Expression |
---|---|
rrxvsca | β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxvsca.r | . . . . 5 β’ β = ( Β·π βπ») | |
2 | rrxvsca.i | . . . . . . 7 β’ (π β πΌ β π) | |
3 | rrxval.r | . . . . . . . 8 β’ π» = (β^βπΌ) | |
4 | 3 | rrxval 25335 | . . . . . . 7 β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
5 | 2, 4 | syl 17 | . . . . . 6 β’ (π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
6 | 5 | fveq2d 6906 | . . . . 5 β’ (π β ( Β·π βπ») = ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))) |
7 | 1, 6 | eqtrid 2780 | . . . 4 β’ (π β β = ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))) |
8 | 7 | oveqd 7443 | . . 3 β’ (π β (π΄ β π) = (π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)) |
9 | 8 | fveq1d 6904 | . 2 β’ (π β ((π΄ β π)βπ½) = ((π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)βπ½)) |
10 | eqid 2728 | . . 3 β’ (βfld freeLMod πΌ) = (βfld freeLMod πΌ) | |
11 | eqid 2728 | . . 3 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(βfld freeLMod πΌ)) | |
12 | rebase 21545 | . . 3 β’ β = (Baseββfld) | |
13 | rrxvsca.a | . . 3 β’ (π β π΄ β β) | |
14 | rrxvsca.x | . . . 4 β’ (π β π β (Baseβπ»)) | |
15 | 5 | fveq2d 6906 | . . . . 5 β’ (π β (Baseβπ») = (Baseβ(toβPreHilβ(βfld freeLMod πΌ)))) |
16 | eqid 2728 | . . . . . 6 β’ (toβPreHilβ(βfld freeLMod πΌ)) = (toβPreHilβ(βfld freeLMod πΌ)) | |
17 | 16, 11 | tcphbas 25167 | . . . . 5 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(toβPreHilβ(βfld freeLMod πΌ))) |
18 | 15, 17 | eqtr4di 2786 | . . . 4 β’ (π β (Baseβπ») = (Baseβ(βfld freeLMod πΌ))) |
19 | 14, 18 | eleqtrd 2831 | . . 3 β’ (π β π β (Baseβ(βfld freeLMod πΌ))) |
20 | rrxvsca.j | . . 3 β’ (π β π½ β πΌ) | |
21 | eqid 2728 | . . . . 5 β’ ( Β·π β(βfld freeLMod πΌ)) = ( Β·π β(βfld freeLMod πΌ)) | |
22 | 16, 21 | tcphvsca 25172 | . . . 4 β’ ( Β·π β(βfld freeLMod πΌ)) = ( Β·π β(toβPreHilβ(βfld freeLMod πΌ))) |
23 | 22 | eqcomi 2737 | . . 3 β’ ( Β·π β(toβPreHilβ(βfld freeLMod πΌ))) = ( Β·π β(βfld freeLMod πΌ)) |
24 | remulr 21550 | . . 3 β’ Β· = (.rββfld) | |
25 | 10, 11, 12, 2, 13, 19, 20, 23, 24 | frlmvscaval 21709 | . 2 β’ (π β ((π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)βπ½) = (π΄ Β· (πβπ½))) |
26 | 9, 25 | eqtrd 2768 | 1 β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcr 11145 Β· cmul 11151 Basecbs 17187 Β·π cvsca 17244 βfldcrefld 21543 freeLMod cfrlm 21687 toβPreHilctcph 25115 β^crrx 25331 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-mulf 11226 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-pws 17438 df-sra 21065 df-rgmod 21066 df-cnfld 21287 df-refld 21544 df-dsmm 21673 df-frlm 21688 df-tng 24513 df-tcph 25117 df-rrx 25333 |
This theorem is referenced by: (None) |
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