rrxplusgvscavalb - Metamath Proof Explorer

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Theorem rrxplusgvscavalb 25523

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 25515	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5	2, 4	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
6	5	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (+_g‘𝐻) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2816	. . . 4 ⊢ (𝜑 → ✚ = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8		rrxplusgvscavalb.r	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐻)
9	5	fveq2d 6886	. . . . . 6 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
10	8, 9	eqtrid 2816	. . . . 5 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
11	10	oveqd 7428	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
12	10	oveqd 7428	. . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌))
13	7, 11, 12	oveq123d 7432	. . 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 2780	. 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 2769	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
16		eqid 2769	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17		rrxplusgvscavalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	5	fveq2d 6886	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
19		rrxbase.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐻)
20		eqid 2769	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
21	20, 16	tcphbas 25347	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
22	18, 19, 21	3eqtr4g 2829	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
23	17, 22	eleqtrd 2871	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
24		rrxplusgvscavalb.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25	24, 22	eleqtrd 2871	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
26		resrng 21740	. . . 4 ⊢ ℝ_fld ∈ *-Ring
27		srngring 20927	. . . 4 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
28	26, 27	mp1i 14	. . 3 ⊢ (𝜑 → ℝ_fld ∈ Ring)
29		rebase 21725	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
30		rrxplusgvscavalb.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31		eqid 2769	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
32	20, 31	tcphvsca 25352	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
33	32	eqcomi 2778	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
34		remulr 21730	. . 3 ⊢ · = (._r‘ℝ_fld)
35		rrxplusgvscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
36	35, 22	eleqtrd 2871	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
37		replusg 21729	. . 3 ⊢ + = (+_g‘ℝ_fld)
38		eqid 2769	. . . . 5 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(ℝ_fld freeLMod 𝐼))
39	20, 38	tchplusg 25348	. . . 4 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
40	39	eqcomi 2778	. . 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 21890	. 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 282	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 209 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ‘cfv 6537 (class class class)co 7411 ℝcr 11099 + caddc 11103 · cmul 11105 Basecbs 17269 +_gcplusg 17310 ·_𝑠 cvsca 17314 Ringcrg 20315 *-Ringcsr 20919 ℝ_fldcrefld 21723 freeLMod cfrlm 21865 toℂPreHilctcph 25295 ℝ^crrx 25511

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179 ax-mulf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-tpos 8222 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-sup 9402 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-rp 13017 df-fz 13536 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-0g 17494 df-prds 17500 df-pws 17502 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-mhm 18841 df-grp 19003 df-minusg 19004 df-sbg 19005 df-subg 19189 df-ghm 19284 df-cmn 19852 df-abl 19853 df-mgp 20217 df-rng 20231 df-ur 20264 df-ring 20317 df-cring 20318 df-oppr 20419 df-dvdsr 20439 df-unit 20440 df-invr 20470 df-dvr 20483 df-rhm 20554 df-subrng 20631 df-subrg 20655 df-drng 20815 df-field 20816 df-staf 20920 df-srng 20921 df-lmod 20961 df-lss 21031 df-sra 21272 df-rgmod 21273 df-cnfld 21492 df-refld 21724 df-dsmm 21851 df-frlm 21866 df-tng 24710 df-tcph 25297 df-rrx 25513

This theorem is referenced by: rrxlinesc 49400 rrxlinec 49401

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