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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 25435 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
6 | 5 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (+g‘𝐻) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
7 | 1, 6 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → ✚ = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
8 | rrxplusgvscavalb.r | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐻) | |
9 | 5 | fveq2d 6911 | . . . . . 6 ⊢ (𝜑 → ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
10 | 8, 9 | eqtrid 2787 | . . . . 5 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
11 | 10 | oveqd 7448 | . . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑋)) |
12 | 10 | oveqd 7448 | . . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))𝑌)) |
13 | 7, 11, 12 | oveq123d 7452 | . . 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 2746 | . 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 2735 | . . 3 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
16 | eqid 2735 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
17 | rrxplusgvscavalb.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | 5 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
19 | rrxbase.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐻) | |
20 | eqid 2735 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
21 | 20, 16 | tcphbas 25267 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
22 | 18, 19, 21 | 3eqtr4g 2800 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
23 | 17, 22 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
24 | rrxplusgvscavalb.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | 24, 22 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
26 | resrng 21657 | . . . 4 ⊢ ℝfld ∈ *-Ring | |
27 | srngring 20864 | . . . 4 ⊢ (ℝfld ∈ *-Ring → ℝfld ∈ Ring) | |
28 | 26, 27 | mp1i 13 | . . 3 ⊢ (𝜑 → ℝfld ∈ Ring) |
29 | rebase 21642 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
30 | rrxplusgvscavalb.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
31 | eqid 2735 | . . . . 5 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) | |
32 | 20, 31 | tcphvsca 25272 | . . . 4 ⊢ ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) = ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
33 | 32 | eqcomi 2744 | . . 3 ⊢ ( ·𝑠 ‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = ( ·𝑠 ‘(ℝfld freeLMod 𝐼)) |
34 | remulr 21647 | . . 3 ⊢ · = (.r‘ℝfld) | |
35 | rrxplusgvscavalb.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
36 | 35, 22 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝfld freeLMod 𝐼))) |
37 | replusg 21646 | . . 3 ⊢ + = (+g‘ℝfld) | |
38 | eqid 2735 | . . . . 5 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(ℝfld freeLMod 𝐼)) | |
39 | 20, 38 | tchplusg 25268 | . . . 4 ⊢ (+g‘(ℝfld freeLMod 𝐼)) = (+g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
40 | 39 | eqcomi 2744 | . . 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 21809 | . 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 279 | 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 206 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 + caddc 11156 · cmul 11158 Basecbs 17245 +gcplusg 17298 ·𝑠 cvsca 17302 Ringcrg 20251 *-Ringcsr 20856 ℝfldcrefld 21640 freeLMod cfrlm 21784 toℂPreHilctcph 25215 ℝ^crrx 25431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-tng 24613 df-tcph 25217 df-rrx 25433 |
This theorem is referenced by: rrxlinesc 48585 rrxlinec 48586 |
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