rrxvsca - Metamath Proof Explorer

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Theorem rrxvsca 25277

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 25270	. . . . . . 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 6889	. . . . 5 ⊢ (𝜑 → ( ·_𝑠 ‘𝐻) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
7	1, 6	eqtrid 2778	. . . 4 ⊢ (𝜑 → ∙ = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
8	7	oveqd 7422	. . 3 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋))
9	8	fveq1d 6887	. 2 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)‘𝐽))
10		eqid 2726	. . 3 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
11		eqid 2726	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
12		rebase 21499	. . 3 ⊢ ℝ = (Base‘ℝ_fld)
13		rrxvsca.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
14		rrxvsca.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻))
15	5	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
16		eqid 2726	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
17	16, 11	tcphbas 25102	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
18	15, 17	eqtr4di 2784	. . . 4 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(ℝ_fld freeLMod 𝐼)))
19	14, 18	eleqtrd 2829	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(ℝ_fld freeLMod 𝐼)))
20		rrxvsca.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
21		eqid 2726	. . . . 5 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
22	16, 21	tcphvsca 25107	. . . 4 ⊢ ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼)) = ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
23	22	eqcomi 2735	. . 3 ⊢ ( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = ( ·_𝑠 ‘(ℝ_fld freeLMod 𝐼))
24		remulr 21504	. . 3 ⊢ · = (._r‘ℝ_fld)
25	10, 11, 12, 2, 13, 19, 20, 23, 24	frlmvscaval 21663	. 2 ⊢ (𝜑 → ((𝐴( ·_𝑠 ‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))
26	9, 25	eqtrd 2766	1 ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 · cmul 11117 Basecbs 17153 ·_𝑠 cvsca 17210 ℝ_fldcrefld 21497 freeLMod cfrlm 21641 toℂPreHilctcph 25050 ℝ^crrx 25266

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-sra 21021 df-rgmod 21022 df-cnfld 21241 df-refld 21498 df-dsmm 21627 df-frlm 21642 df-tng 24448 df-tcph 25052 df-rrx 25268

This theorem is referenced by: (None)

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