rrxplusgvscavalb - Metamath Proof Explorer

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Theorem rrxplusgvscavalb 25375

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
rrxplusgvscavalb	⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))

Distinct variable groups: 𝑖,𝐼 𝐴,𝑖 𝐶,𝑖 𝑖,𝑋 𝑖,𝑌 𝑖,𝑍 𝜑,𝑖 Allowed substitution hints: 𝐵(𝑖) ✚ (𝑖) ∙ (𝑖) 𝐻(𝑖) 𝑉(𝑖)

**Proof of Theorem rrxplusgvscavalb**
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 25367	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5	2, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
6	5	fveq2d 6839	. . . . 5 ⊢ (𝜑 → (+_g‘𝐻) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2784	. . . 4 ⊢ (𝜑 → ✚ = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8		rrxplusgvscavalb.r	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐻)
9	5	fveq2d 6839	. . . . . 6 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
10	8, 9	eqtrid 2784	. . . . 5 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
11	10	oveqd 7378	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
12	10	oveqd 7378	. . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌))
13	7, 11, 12	oveq123d 7382	. . 3 ⊢ (𝜑 → ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) = ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)(+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))(𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌)))
14	13	eqeq2d 2748	. 2 ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ 𝑍 = ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)(+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))(𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌))))
15		eqid 2737	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
16		eqid 2737	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17		rrxplusgvscavalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	5	fveq2d 6839	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
19		rrxbase.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐻)
20		eqid 2737	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
21	20, 16	tcphbas 25199	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
22	18, 19, 21	3eqtr4g 2797	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
23	17, 22	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
24		rrxplusgvscavalb.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25	24, 22	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
26		resrng 21614	. . . 4 ⊢ ℝ_fld ∈ *-Ring
27		srngring 20817	. . . 4 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
28	26, 27	mp1i 13	. . 3 ⊢ (𝜑 → ℝ_fld ∈ Ring)
29		rebase 21599	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
30		rrxplusgvscavalb.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31		eqid 2737	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
32	20, 31	tcphvsca 25204	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
33	32	eqcomi 2746	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
34		remulr 21604	. . 3 ⊢ · = (._r‘ℝ_fld)
35		rrxplusgvscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
36	35, 22	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
37		replusg 21603	. . 3 ⊢ + = (+_g‘ℝ_fld)
38		eqid 2737	. . . . 5 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(ℝ_fld freeLMod 𝐼))
39	20, 38	tchplusg 25200	. . . 4 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
40	39	eqcomi 2746	. . 3 ⊢ (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (+_g‘(ℝ_fld 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 21764	. 2 ⊢ (𝜑 → (𝑍 = ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)(+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))(𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld 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 1542 ∈ wcel 2114 ∀wral 3052 ‘cfv 6493 (class class class)co 7361 ℝcr 11031 + caddc 11035 · cmul 11037 Basecbs 17173 +_gcplusg 17214 ·_𝑠 cvsca 17218 Ringcrg 20208 *-Ringcsr 20809 ℝ_fldcrefld 21597 freeLMod cfrlm 21739 toℂPreHilctcph 25147 ℝ^crrx 25363

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111 ax-mulf 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-rp 12937 df-fz 13456 df-seq 13958 df-exp 14018 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-0g 17398 df-prds 17404 df-pws 17406 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-ghm 19182 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-subrng 20517 df-subrg 20541 df-drng 20702 df-field 20703 df-staf 20810 df-srng 20811 df-lmod 20851 df-lss 20921 df-sra 21163 df-rgmod 21164 df-cnfld 21348 df-refld 21598 df-dsmm 21725 df-frlm 21740 df-tng 24562 df-tcph 25149 df-rrx 25365

This theorem is referenced by: rrxlinesc 49226 rrxlinec 49227

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