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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 23985 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
6 | 5 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (+g‘𝐻) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
7 | 1, 6 | syl5eq 2867 | . . . 4 ⊢ (𝜑 → ✚ = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
8 | rrxplusgvscavalb.r | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐻) | |
9 | 5 | fveq2d 6667 | . . . . . 6 ⊢ (𝜑 → ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
10 | 8, 9 | syl5eq 2867 | . . . . 5 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
11 | 10 | oveqd 7166 | . . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)) |
12 | 10 | oveqd 7166 | . . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)) |
13 | 7, 11, 12 | oveq123d 7170 | . . 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 2831 | . 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 2820 | . . 3 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
16 | eqid 2820 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
17 | rrxplusgvscavalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | 5 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
19 | rrxbase.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐻) | |
20 | eqid 2820 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
21 | 20, 16 | tcphbas 23817 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
22 | 18, 19, 21 | 3eqtr4g 2880 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
23 | 17, 22 | eleqtrd 2914 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
24 | rrxplusgvscavalb.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | 24, 22 | eleqtrd 2914 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
26 | recrng 20760 | . . . 4 ⊢ ℝfld ∈ *-Ring | |
27 | srngring 19618 | . . . 4 ⊢ (ℝfld ∈ *-Ring → ℝfld ∈ Ring) | |
28 | 26, 27 | mp1i 13 | . . 3 ⊢ (𝜑 → ℝfld ∈ Ring) |
29 | rebase 20745 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
30 | rrxplusgvscavalb.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
31 | eqid 2820 | . . . . 5 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) | |
32 | 20, 31 | tcphvsca 23822 | . . . 4 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
33 | 32 | eqcomi 2829 | . . 3 ⊢ ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) |
34 | remulr 20750 | . . 3 ⊢ · = (.r‘ℝfld) | |
35 | rrxplusgvscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
36 | 35, 22 | eleqtrd 2914 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
37 | replusg 20749 | . . 3 ⊢ + = (+g‘ℝfld) | |
38 | eqid 2820 | . . . . 5 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(ℝfld freeLMod 𝐼)) | |
39 | 20, 38 | tchplusg 23818 | . . . 4 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
40 | 39 | eqcomi 2829 | . . 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 20910 | . 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 281 | 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 208 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ‘cfv 6348 (class class class)co 7149 ℝcr 10529 + caddc 10533 · cmul 10535 Basecbs 16478 +gcplusg 16560 ·𝑠 cvsca 16564 Ringcrg 19292 *-Ringcsr 19610 ℝfldcrefld 20743 freeLMod cfrlm 20885 toℂPreHilctcph 23766 ℝ^crrx 23981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-seq 13367 df-exp 13427 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-0g 16710 df-prds 16716 df-pws 16718 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-ghm 18351 df-cmn 18903 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-rnghom 19462 df-drng 19499 df-field 19500 df-subrg 19528 df-staf 19611 df-srng 19612 df-lmod 19631 df-lss 19699 df-sra 19939 df-rgmod 19940 df-cnfld 20541 df-refld 20744 df-dsmm 20871 df-frlm 20886 df-tng 23189 df-tcph 23768 df-rrx 23983 |
This theorem is referenced by: rrxlinesc 44796 rrxlinec 44797 |
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