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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 | ⊢ 𝐻 = (ℝ^‘𝐼) |
rrxbase.b | ⊢ 𝐵 = (Base‘𝐻) |
rrxplusgvscavalb.r | ⊢ ∙ = ( ·𝑠 ‘𝐻) |
rrxplusgvscavalb.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
rrxplusgvscavalb.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rrxplusgvscavalb.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rrxplusgvscavalb.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rrxplusgvscavalb.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
rrxplusgvscavalb.p | ⊢ ✚ = (+g‘𝐻) |
rrxplusgvscavalb.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
rrxplusgvscavalb | ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxplusgvscavalb.p | . . . . 5 ⊢ ✚ = (+g‘𝐻) | |
2 | rrxplusgvscavalb.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | rrxval.r | . . . . . . . 8 ⊢ 𝐻 = (ℝ^‘𝐼) | |
4 | 3 | rrxval 23991 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
6 | 5 | fveq2d 6649 | . . . . 5 ⊢ (𝜑 → (+g‘𝐻) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
7 | 1, 6 | syl5eq 2845 | . . . 4 ⊢ (𝜑 → ✚ = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
8 | rrxplusgvscavalb.r | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐻) | |
9 | 5 | fveq2d 6649 | . . . . . 6 ⊢ (𝜑 → ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
10 | 8, 9 | syl5eq 2845 | . . . . 5 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
11 | 10 | oveqd 7152 | . . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)) |
12 | 10 | oveqd 7152 | . . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)) |
13 | 7, 11, 12 | oveq123d 7156 | . . 3 ⊢ (𝜑 → ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) = ((𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)(+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))(𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌))) |
14 | 13 | eqeq2d 2809 | . 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ 𝑍 = ((𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)(+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))(𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)))) |
15 | eqid 2798 | . . 3 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
16 | eqid 2798 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
17 | rrxplusgvscavalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | 5 | fveq2d 6649 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
19 | rrxbase.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐻) | |
20 | eqid 2798 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
21 | 20, 16 | tcphbas 23823 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
22 | 18, 19, 21 | 3eqtr4g 2858 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
23 | 17, 22 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
24 | rrxplusgvscavalb.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | 24, 22 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
26 | recrng 20310 | . . . 4 ⊢ ℝfld ∈ *-Ring | |
27 | srngring 19616 | . . . 4 ⊢ (ℝfld ∈ *-Ring → ℝfld ∈ Ring) | |
28 | 26, 27 | mp1i 13 | . . 3 ⊢ (𝜑 → ℝfld ∈ Ring) |
29 | rebase 20295 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
30 | rrxplusgvscavalb.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
31 | eqid 2798 | . . . . 5 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) | |
32 | 20, 31 | tcphvsca 23828 | . . . 4 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
33 | 32 | eqcomi 2807 | . . 3 ⊢ ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) |
34 | remulr 20300 | . . 3 ⊢ · = (.r‘ℝfld) | |
35 | rrxplusgvscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
36 | 35, 22 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
37 | replusg 20299 | . . 3 ⊢ + = (+g‘ℝfld) | |
38 | eqid 2798 | . . . . 5 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(ℝfld freeLMod 𝐼)) | |
39 | 20, 38 | tchplusg 23824 | . . . 4 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
40 | 39 | eqcomi 2807 | . . 3 ⊢ (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = (+g‘(ℝfld freeLMod 𝐼)) |
41 | rrxplusgvscavalb.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
42 | 15, 16, 2, 23, 25, 28, 29, 30, 33, 34, 36, 37, 40, 41 | frlmvplusgscavalb 20460 | . 2 ⊢ (𝜑 → (𝑍 = ((𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)(+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))(𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
43 | 14, 42 | bitrd 282 | 1 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 + caddc 10529 · cmul 10531 Basecbs 16475 +gcplusg 16557 ·𝑠 cvsca 16561 Ringcrg 19290 *-Ringcsr 19608 ℝfldcrefld 20293 freeLMod cfrlm 20435 toℂPreHilctcph 23772 ℝ^crrx 23987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-tng 23191 df-tcph 23774 df-rrx 23989 |
This theorem is referenced by: rrxlinesc 45149 rrxlinec 45150 |
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