rrxplusgvscavalb - Metamath Proof Explorer

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Theorem rrxplusgvscavalb 25310

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 25302	. . . . . . 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 6895	. . . . 5 ⊢ (𝜑 → (+_g‘𝐻) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2779	. . . 4 ⊢ (𝜑 → ✚ = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8		rrxplusgvscavalb.r	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐻)
9	5	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
10	8, 9	eqtrid 2779	. . . . 5 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
11	10	oveqd 7431	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
12	10	oveqd 7431	. . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌))
13	7, 11, 12	oveq123d 7435	. . 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 2738	. 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 2727	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
16		eqid 2727	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17		rrxplusgvscavalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	5	fveq2d 6895	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
19		rrxbase.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐻)
20		eqid 2727	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
21	20, 16	tcphbas 25134	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
22	18, 19, 21	3eqtr4g 2792	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
23	17, 22	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
24		rrxplusgvscavalb.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25	24, 22	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
26		resrng 21540	. . . 4 ⊢ ℝ_fld ∈ *-Ring
27		srngring 20721	. . . 4 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
28	26, 27	mp1i 13	. . 3 ⊢ (𝜑 → ℝ_fld ∈ Ring)
29		rebase 21525	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
30		rrxplusgvscavalb.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31		eqid 2727	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
32	20, 31	tcphvsca 25139	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
33	32	eqcomi 2736	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
34		remulr 21530	. . 3 ⊢ · = (._r‘ℝ_fld)
35		rrxplusgvscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
36	35, 22	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
37		replusg 21529	. . 3 ⊢ + = (+_g‘ℝ_fld)
38		eqid 2727	. . . . 5 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(ℝ_fld freeLMod 𝐼))
39	20, 38	tchplusg 25135	. . . 4 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
40	39	eqcomi 2736	. . 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 21692	. 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 205 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 + caddc 11133 · cmul 11135 Basecbs 17171 +_gcplusg 17224 ·_𝑠 cvsca 17228 Ringcrg 20164 *-Ringcsr 20713 ℝ_fldcrefld 21523 freeLMod cfrlm 21667 toℂPreHilctcph 25082 ℝ^crrx 25298

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-rp 12999 df-fz 13509 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-prds 17420 df-pws 17422 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-ghm 19159 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-field 20616 df-staf 20714 df-srng 20715 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-cnfld 21267 df-refld 21524 df-dsmm 21653 df-frlm 21668 df-tng 24480 df-tcph 25084 df-rrx 25300

This theorem is referenced by: rrxlinesc 47731 rrxlinec 47732

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