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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 23993 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
6 | 5 | fveq2d 6677 | . . . . 5 ⊢ (𝜑 → (+g‘𝐻) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
7 | 1, 6 | syl5eq 2871 | . . . 4 ⊢ (𝜑 → ✚ = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
8 | rrxplusgvscavalb.r | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐻) | |
9 | 5 | fveq2d 6677 | . . . . . 6 ⊢ (𝜑 → ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
10 | 8, 9 | syl5eq 2871 | . . . . 5 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
11 | 10 | oveqd 7176 | . . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)) |
12 | 10 | oveqd 7176 | . . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)) |
13 | 7, 11, 12 | oveq123d 7180 | . . 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 2835 | . 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 2824 | . . 3 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
16 | eqid 2824 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
17 | rrxplusgvscavalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | 5 | fveq2d 6677 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
19 | rrxbase.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐻) | |
20 | eqid 2824 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
21 | 20, 16 | tcphbas 23825 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
22 | 18, 19, 21 | 3eqtr4g 2884 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
23 | 17, 22 | eleqtrd 2918 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
24 | rrxplusgvscavalb.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | 24, 22 | eleqtrd 2918 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
26 | recrng 20768 | . . . 4 ⊢ ℝfld ∈ *-Ring | |
27 | srngring 19626 | . . . 4 ⊢ (ℝfld ∈ *-Ring → ℝfld ∈ Ring) | |
28 | 26, 27 | mp1i 13 | . . 3 ⊢ (𝜑 → ℝfld ∈ Ring) |
29 | rebase 20753 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
30 | rrxplusgvscavalb.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
31 | eqid 2824 | . . . . 5 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) | |
32 | 20, 31 | tcphvsca 23830 | . . . 4 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
33 | 32 | eqcomi 2833 | . . 3 ⊢ ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) |
34 | remulr 20758 | . . 3 ⊢ · = (.r‘ℝfld) | |
35 | rrxplusgvscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
36 | 35, 22 | eleqtrd 2918 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
37 | replusg 20757 | . . 3 ⊢ + = (+g‘ℝfld) | |
38 | eqid 2824 | . . . . 5 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(ℝfld freeLMod 𝐼)) | |
39 | 20, 38 | tchplusg 23826 | . . . 4 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
40 | 39 | eqcomi 2833 | . . 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 20918 | . 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 3141 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 + caddc 10543 · cmul 10545 Basecbs 16486 +gcplusg 16568 ·𝑠 cvsca 16572 Ringcrg 19300 *-Ringcsr 19618 ℝfldcrefld 20751 freeLMod cfrlm 20893 toℂPreHilctcph 23774 ℝ^crrx 23989 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-0g 16718 df-prds 16724 df-pws 16726 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-ghm 18359 df-cmn 18911 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-field 19508 df-subrg 19536 df-staf 19619 df-srng 19620 df-lmod 19639 df-lss 19707 df-sra 19947 df-rgmod 19948 df-cnfld 20549 df-refld 20752 df-dsmm 20879 df-frlm 20894 df-tng 23197 df-tcph 23776 df-rrx 23991 |
This theorem is referenced by: rrxlinesc 44729 rrxlinec 44730 |
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