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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 25359 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
6 | 5 | fveq2d 6900 | . . . . 5 ⊢ (𝜑 → (+g‘𝐻) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
7 | 1, 6 | eqtrid 2777 | . . . 4 ⊢ (𝜑 → ✚ = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
8 | rrxplusgvscavalb.r | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐻) | |
9 | 5 | fveq2d 6900 | . . . . . 6 ⊢ (𝜑 → ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
10 | 8, 9 | eqtrid 2777 | . . . . 5 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
11 | 10 | oveqd 7436 | . . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)) |
12 | 10 | oveqd 7436 | . . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)) |
13 | 7, 11, 12 | oveq123d 7440 | . . 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 2736 | . 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 2725 | . . 3 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
16 | eqid 2725 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
17 | rrxplusgvscavalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | 5 | fveq2d 6900 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
19 | rrxbase.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐻) | |
20 | eqid 2725 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
21 | 20, 16 | tcphbas 25191 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
22 | 18, 19, 21 | 3eqtr4g 2790 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
23 | 17, 22 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
24 | rrxplusgvscavalb.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | 24, 22 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
26 | resrng 21570 | . . . 4 ⊢ ℝfld ∈ *-Ring | |
27 | srngring 20744 | . . . 4 ⊢ (ℝfld ∈ *-Ring → ℝfld ∈ Ring) | |
28 | 26, 27 | mp1i 13 | . . 3 ⊢ (𝜑 → ℝfld ∈ Ring) |
29 | rebase 21555 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
30 | rrxplusgvscavalb.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
31 | eqid 2725 | . . . . 5 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) | |
32 | 20, 31 | tcphvsca 25196 | . . . 4 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
33 | 32 | eqcomi 2734 | . . 3 ⊢ ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) |
34 | remulr 21560 | . . 3 ⊢ · = (.r‘ℝfld) | |
35 | rrxplusgvscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
36 | 35, 22 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
37 | replusg 21559 | . . 3 ⊢ + = (+g‘ℝfld) | |
38 | eqid 2725 | . . . . 5 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(ℝfld freeLMod 𝐼)) | |
39 | 20, 38 | tchplusg 25192 | . . . 4 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
40 | 39 | eqcomi 2734 | . . 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 21722 | . 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 278 | 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 205 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 + caddc 11143 · cmul 11145 Basecbs 17183 +gcplusg 17236 ·𝑠 cvsca 17240 Ringcrg 20185 *-Ringcsr 20736 ℝfldcrefld 21553 freeLMod cfrlm 21697 toℂPreHilctcph 25139 ℝ^crrx 25355 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-0g 17426 df-prds 17432 df-pws 17434 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-ghm 19176 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrng 20495 df-subrg 20520 df-drng 20638 df-field 20639 df-staf 20737 df-srng 20738 df-lmod 20757 df-lss 20828 df-sra 21070 df-rgmod 21071 df-cnfld 21297 df-refld 21554 df-dsmm 21683 df-frlm 21698 df-tng 24537 df-tcph 25141 df-rrx 25357 |
This theorem is referenced by: rrxlinesc 47994 rrxlinec 47995 |
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