rrxplusgvscavalb - Metamath Proof Explorer

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Theorem rrxplusgvscavalb 25351

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 25343	. . . . . . 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 6838	. . . . 5 ⊢ (𝜑 → (+_g‘𝐻) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2783	. . . 4 ⊢ (𝜑 → ✚ = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8		rrxplusgvscavalb.r	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐻)
9	5	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
10	8, 9	eqtrid 2783	. . . . 5 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
11	10	oveqd 7375	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
12	10	oveqd 7375	. . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌))
13	7, 11, 12	oveq123d 7379	. . 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 2747	. 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 2736	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
16		eqid 2736	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17		rrxplusgvscavalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	5	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
19		rrxbase.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐻)
20		eqid 2736	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
21	20, 16	tcphbas 25175	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
22	18, 19, 21	3eqtr4g 2796	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
23	17, 22	eleqtrd 2838	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
24		rrxplusgvscavalb.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25	24, 22	eleqtrd 2838	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
26		resrng 21576	. . . 4 ⊢ ℝ_fld ∈ *-Ring
27		srngring 20779	. . . 4 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
28	26, 27	mp1i 13	. . 3 ⊢ (𝜑 → ℝ_fld ∈ Ring)
29		rebase 21561	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
30		rrxplusgvscavalb.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31		eqid 2736	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
32	20, 31	tcphvsca 25180	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
33	32	eqcomi 2745	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
34		remulr 21566	. . 3 ⊢ · = (._r‘ℝ_fld)
35		rrxplusgvscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
36	35, 22	eleqtrd 2838	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
37		replusg 21565	. . 3 ⊢ + = (+_g‘ℝ_fld)
38		eqid 2736	. . . . 5 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(ℝ_fld freeLMod 𝐼))
39	20, 38	tchplusg 25176	. . . 4 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
40	39	eqcomi 2745	. . 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 21726	. 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 1541 ∈ wcel 2113 ∀wral 3051 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 + caddc 11029 · cmul 11031 Basecbs 17136 +_gcplusg 17177 ·_𝑠 cvsca 17181 Ringcrg 20168 *-Ringcsr 20771 ℝ_fldcrefld 21559 freeLMod cfrlm 21701 toℂPreHilctcph 25123 ℝ^crrx 25339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-fz 13424 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-0g 17361 df-prds 17367 df-pws 17369 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-ghm 19142 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-rhm 20408 df-subrng 20479 df-subrg 20503 df-drng 20664 df-field 20665 df-staf 20772 df-srng 20773 df-lmod 20813 df-lss 20883 df-sra 21125 df-rgmod 21126 df-cnfld 21310 df-refld 21560 df-dsmm 21687 df-frlm 21702 df-tng 24528 df-tcph 25125 df-rrx 25341

This theorem is referenced by: rrxlinesc 48981 rrxlinec 48982

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