rrxvsca - Metamath Proof Explorer

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Theorem rrxvsca 25383

Description: The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.)

**Hypotheses**
Ref	Expression
rrxval.r	⊢ 𝐻 = (ℝ^‘𝐼)
rrxbase.b	⊢ 𝐵 = (Base‘𝐻)
rrxvsca.r	⊢ ∙ = ( ·_𝑠 ‘𝐻)
rrxvsca.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
rrxvsca.j	⊢ (𝜑 → 𝐽 ∈ 𝐼)
rrxvsca.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
rrxvsca.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))

**Assertion**
Ref	Expression
rrxvsca	⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))

**Proof of Theorem rrxvsca**
Step	Hyp	Ref	Expression
1		rrxvsca.r	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘𝐻)
2		rrxvsca.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
3		rrxval.r	. . . . . . . 8 ⊢ 𝐻 = (ℝ^‘𝐼)
4	3	rrxval 25376	. . . . . . 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 6835	. . . . 5 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2788	. . . 4 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8	7	oveqd 7377	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
9	8	fveq1d 6833	. 2 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)‘𝐽))
10		eqid 2741	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
11		eqid 2741	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
12		rebase 21585	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
13		rrxvsca.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
14		rrxvsca.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15	5	fveq2d 6835	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
16		eqid 2741	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
17	16, 11	tcphbas 25208	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
18	15, 17	eqtr4di 2794	. . . 4 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(ℝ_fld freeLMod 𝐼)))
19	14, 18	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
20		rrxvsca.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
21		eqid 2741	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
22	16, 21	tcphvsca 25213	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
23	22	eqcomi 2750	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
24		remulr 21590	. . 3 ⊢ · = (._r‘ℝ_fld)
25	10, 11, 12, 2, 13, 19, 20, 23, 24	frlmvscaval 21747	. 2 ⊢ (𝜑 → ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))
26	9, 25	eqtrd 2776	1 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1548 ∈ wcel 2121 ‘cfv 6489 (class class class)co 7360 ℝcr 11032 · cmul 11038 Basecbs 17174 ·_𝑠 cvsca 17219 ℝ_fldcrefld 21583 freeLMod cfrlm 21725 toℂPreHilctcph 25156 ℝ^crrx 25372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 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 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-prds 17405 df-pws 17407 df-sra 21167 df-rgmod 21168 df-cnfld 21352 df-refld 21584 df-dsmm 21711 df-frlm 21726 df-tng 24571 df-tcph 25158 df-rrx 25374

This theorem is referenced by: (None)

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