rrxplusgvscavalb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rrxplusgvscavalb		Structured version Visualization version GIF version

Theorem rrxplusgvscavalb 25353

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 25345	. . . . . . 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 6898	. . . . 5 ⊢ (𝜑 → (+_g‘𝐻) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2777	. . . 4 ⊢ (𝜑 → ✚ = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8		rrxplusgvscavalb.r	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐻)
9	5	fveq2d 6898	. . . . . 6 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
10	8, 9	eqtrid 2777	. . . . 5 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
11	10	oveqd 7434	. . . 4 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
12	10	oveqd 7434	. . . 4 ⊢ (𝜑 → (𝐶 ∙ 𝑌) = (𝐶( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑌))
13	7, 11, 12	oveq123d 7438	. . 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 2736	. 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 2725	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
16		eqid 2725	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
17		rrxplusgvscavalb.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	5	fveq2d 6898	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
19		rrxbase.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐻)
20		eqid 2725	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
21	20, 16	tcphbas 25177	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
22	18, 19, 21	3eqtr4g 2790	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
23	17, 22	eleqtrd 2827	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
24		rrxplusgvscavalb.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25	24, 22	eleqtrd 2827	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
26		resrng 21557	. . . 4 ⊢ ℝ_fld ∈ *-Ring
27		srngring 20736	. . . 4 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
28	26, 27	mp1i 13	. . 3 ⊢ (𝜑 → ℝ_fld ∈ Ring)
29		rebase 21542	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
30		rrxplusgvscavalb.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31		eqid 2725	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
32	20, 31	tcphvsca 25182	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
33	32	eqcomi 2734	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
34		remulr 21547	. . 3 ⊢ · = (._r‘ℝ_fld)
35		rrxplusgvscavalb.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
36	35, 22	eleqtrd 2827	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
37		replusg 21546	. . 3 ⊢ + = (+_g‘ℝ_fld)
38		eqid 2725	. . . . 5 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(ℝ_fld freeLMod 𝐼))
39	20, 38	tchplusg 25178	. . . 4 ⊢ (+_g‘(ℝ_fld freeLMod 𝐼)) = (+_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
40	39	eqcomi 2734	. . 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 21709	. 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 278	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 1533 ∈ wcel 2098 ∀wral 3051 ‘cfv 6547 (class class class)co 7417 ℝcr 11137 + caddc 11141 · cmul 11143 Basecbs 17179 +_gcplusg 17232 ·_𝑠 cvsca 17236 Ringcrg 20177 *-Ringcsr 20728 ℝ_fldcrefld 21540 freeLMod cfrlm 21684 toℂPreHilctcph 25125 ℝ^crrx 25341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-ghm 19172 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-drng 20630 df-field 20631 df-staf 20729 df-srng 20730 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-cnfld 21284 df-refld 21541 df-dsmm 21670 df-frlm 21685 df-tng 24523 df-tcph 25127 df-rrx 25343

This theorem is referenced by: rrxlinesc 47920 rrxlinec 47921

W3C validator