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Mirrors > Home > MPE Home > Th. List > rrxplusgvscavalb | Structured version Visualization version GIF version |
Description: The result of the addition combined with scalar multiplication in a generalized Euclidean space is defined by its coordinate-wise operations. (Contributed by AV, 21-Jan-2023.) |
Ref | Expression |
---|---|
rrxval.r | β’ π» = (β^βπΌ) |
rrxbase.b | β’ π΅ = (Baseβπ») |
rrxplusgvscavalb.r | β’ β = ( Β·π βπ») |
rrxplusgvscavalb.i | β’ (π β πΌ β π) |
rrxplusgvscavalb.a | β’ (π β π΄ β β) |
rrxplusgvscavalb.x | β’ (π β π β π΅) |
rrxplusgvscavalb.y | β’ (π β π β π΅) |
rrxplusgvscavalb.z | β’ (π β π β π΅) |
rrxplusgvscavalb.p | β’ β = (+gβπ») |
rrxplusgvscavalb.c | β’ (π β πΆ β β) |
Ref | Expression |
---|---|
rrxplusgvscavalb | β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxplusgvscavalb.p | . . . . 5 β’ β = (+gβπ») | |
2 | rrxplusgvscavalb.i | . . . . . . 7 β’ (π β πΌ β π) | |
3 | rrxval.r | . . . . . . . 8 β’ π» = (β^βπΌ) | |
4 | 3 | rrxval 24754 | . . . . . . 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 6847 | . . . . 5 β’ (π β (+gβπ») = (+gβ(toβPreHilβ(βfld freeLMod πΌ)))) |
7 | 1, 6 | eqtrid 2789 | . . . 4 β’ (π β β = (+gβ(toβPreHilβ(βfld freeLMod πΌ)))) |
8 | rrxplusgvscavalb.r | . . . . . 6 β’ β = ( Β·π βπ») | |
9 | 5 | fveq2d 6847 | . . . . . 6 β’ (π β ( Β·π βπ») = ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))) |
10 | 8, 9 | eqtrid 2789 | . . . . 5 β’ (π β β = ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))) |
11 | 10 | oveqd 7375 | . . . 4 β’ (π β (π΄ β π) = (π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)) |
12 | 10 | oveqd 7375 | . . . 4 β’ (π β (πΆ β π) = (πΆ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)) |
13 | 7, 11, 12 | oveq123d 7379 | . . 3 β’ (π β ((π΄ β π) β (πΆ β π)) = ((π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)(+gβ(toβPreHilβ(βfld freeLMod πΌ)))(πΆ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π))) |
14 | 13 | eqeq2d 2748 | . 2 β’ (π β (π = ((π΄ β π) β (πΆ β π)) β π = ((π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)(+gβ(toβPreHilβ(βfld freeLMod πΌ)))(πΆ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)))) |
15 | eqid 2737 | . . 3 β’ (βfld freeLMod πΌ) = (βfld freeLMod πΌ) | |
16 | eqid 2737 | . . 3 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(βfld freeLMod πΌ)) | |
17 | rrxplusgvscavalb.x | . . . 4 β’ (π β π β π΅) | |
18 | 5 | fveq2d 6847 | . . . . 5 β’ (π β (Baseβπ») = (Baseβ(toβPreHilβ(βfld freeLMod πΌ)))) |
19 | rrxbase.b | . . . . 5 β’ π΅ = (Baseβπ») | |
20 | eqid 2737 | . . . . . 6 β’ (toβPreHilβ(βfld freeLMod πΌ)) = (toβPreHilβ(βfld freeLMod πΌ)) | |
21 | 20, 16 | tcphbas 24586 | . . . . 5 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(toβPreHilβ(βfld freeLMod πΌ))) |
22 | 18, 19, 21 | 3eqtr4g 2802 | . . . 4 β’ (π β π΅ = (Baseβ(βfld freeLMod πΌ))) |
23 | 17, 22 | eleqtrd 2840 | . . 3 β’ (π β π β (Baseβ(βfld freeLMod πΌ))) |
24 | rrxplusgvscavalb.z | . . . 4 β’ (π β π β π΅) | |
25 | 24, 22 | eleqtrd 2840 | . . 3 β’ (π β π β (Baseβ(βfld freeLMod πΌ))) |
26 | resrng 21028 | . . . 4 β’ βfld β *-Ring | |
27 | srngring 20314 | . . . 4 β’ (βfld β *-Ring β βfld β Ring) | |
28 | 26, 27 | mp1i 13 | . . 3 β’ (π β βfld β Ring) |
29 | rebase 21013 | . . 3 β’ β = (Baseββfld) | |
30 | rrxplusgvscavalb.a | . . 3 β’ (π β π΄ β β) | |
31 | eqid 2737 | . . . . 5 β’ ( Β·π β(βfld freeLMod πΌ)) = ( Β·π β(βfld freeLMod πΌ)) | |
32 | 20, 31 | tcphvsca 24591 | . . . 4 β’ ( Β·π β(βfld freeLMod πΌ)) = ( Β·π β(toβPreHilβ(βfld freeLMod πΌ))) |
33 | 32 | eqcomi 2746 | . . 3 β’ ( Β·π β(toβPreHilβ(βfld freeLMod πΌ))) = ( Β·π β(βfld freeLMod πΌ)) |
34 | remulr 21018 | . . 3 β’ Β· = (.rββfld) | |
35 | rrxplusgvscavalb.y | . . . 4 β’ (π β π β π΅) | |
36 | 35, 22 | eleqtrd 2840 | . . 3 β’ (π β π β (Baseβ(βfld freeLMod πΌ))) |
37 | replusg 21017 | . . 3 β’ + = (+gββfld) | |
38 | eqid 2737 | . . . . 5 β’ (+gβ(βfld freeLMod πΌ)) = (+gβ(βfld freeLMod πΌ)) | |
39 | 20, 38 | tchplusg 24587 | . . . 4 β’ (+gβ(βfld freeLMod πΌ)) = (+gβ(toβPreHilβ(βfld freeLMod πΌ))) |
40 | 39 | eqcomi 2746 | . . 3 β’ (+gβ(toβPreHilβ(βfld freeLMod πΌ))) = (+gβ(βfld freeLMod πΌ)) |
41 | rrxplusgvscavalb.c | . . 3 β’ (π β πΆ β β) | |
42 | 15, 16, 2, 23, 25, 28, 29, 30, 33, 34, 36, 37, 40, 41 | frlmvplusgscavalb 21180 | . 2 β’ (π β (π = ((π΄( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)(+gβ(toβPreHilβ(βfld freeLMod πΌ)))(πΆ( Β·π β(toβPreHilβ(βfld freeLMod πΌ)))π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
43 | 14, 42 | bitrd 279 | 1 β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 βwral 3065 βcfv 6497 (class class class)co 7358 βcr 11051 + caddc 11055 Β· cmul 11057 Basecbs 17084 +gcplusg 17134 Β·π cvsca 17138 Ringcrg 19965 *-Ringcsr 20306 βfldcrefld 21011 freeLMod cfrlm 21155 toβPreHilctcph 24534 β^crrx 24750 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9307 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-rp 12917 df-fz 13426 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-hom 17158 df-cco 17159 df-0g 17324 df-prds 17330 df-pws 17332 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-ghm 19007 df-cmn 19565 df-mgp 19898 df-ur 19915 df-ring 19967 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-rnghom 20147 df-drng 20188 df-field 20189 df-subrg 20223 df-staf 20307 df-srng 20308 df-lmod 20327 df-lss 20396 df-sra 20636 df-rgmod 20637 df-cnfld 20800 df-refld 21012 df-dsmm 21141 df-frlm 21156 df-tng 23943 df-tcph 24536 df-rrx 24752 |
This theorem is referenced by: rrxlinesc 46828 rrxlinec 46829 |
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