rrxprds - Metamath Proof Explorer

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Theorem rrxprds 24906

Description: Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)

**Hypotheses**
Ref	Expression
rrxval.r	⊢ 𝐻 = (ℝ^‘𝐼)
rrxbase.b	⊢ 𝐵 = (Base‘𝐻)

**Assertion**
Ref	Expression
rrxprds	⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵)))

**Proof of Theorem rrxprds**
Step	Hyp	Ref	Expression
1		rrxval.r	. . 3 ⊢ 𝐻 = (ℝ^‘𝐼)
2	1	rrxval 24904	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		refld 21172	. . . . 5 ⊢ ℝ_fld ∈ Field
4		eqid 2733	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5		eqid 2733	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6	4, 5	frlmpws 21305	. . . . 5 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) = (((ringLMod‘ℝ_fld) ↑_s 𝐼) ↾_s (Base‘(ℝ_fld freeLMod 𝐼))))
7	3, 6	mpan 689	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) = (((ringLMod‘ℝ_fld) ↑_s 𝐼) ↾_s (Base‘(ℝ_fld freeLMod 𝐼))))
8		fvex 6905	. . . . . . 7 ⊢ ((subringAlg ‘ℝ_fld)‘ℝ) ∈ V
9		rlmval 20813	. . . . . . . . . 10 ⊢ (ringLMod‘ℝ_fld) = ((subringAlg ‘ℝ_fld)‘(Base‘ℝ_fld))
10		rebase 21159	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
11	10	fveq2i 6895	. . . . . . . . . 10 ⊢ ((subringAlg ‘ℝ_fld)‘ℝ) = ((subringAlg ‘ℝ_fld)‘(Base‘ℝ_fld))
12	9, 11	eqtr4i 2764	. . . . . . . . 9 ⊢ (ringLMod‘ℝ_fld) = ((subringAlg ‘ℝ_fld)‘ℝ)
13	12	oveq1i 7419	. . . . . . . 8 ⊢ ((ringLMod‘ℝ_fld) ↑_s 𝐼) = (((subringAlg ‘ℝ_fld)‘ℝ) ↑_s 𝐼)
14	10	ressid 17189	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ Field → (ℝ_fld ↾_s ℝ) = ℝ_fld)
15	3, 14	ax-mp 5	. . . . . . . . 9 ⊢ (ℝ_fld ↾_s ℝ) = ℝ_fld
16		eqidd 2734	. . . . . . . . . . 11 ⊢ (⊤ → ((subringAlg ‘ℝ_fld)‘ℝ) = ((subringAlg ‘ℝ_fld)‘ℝ))
17	10	eqimssi 4043	. . . . . . . . . . . 12 ⊢ ℝ ⊆ (Base‘ℝ_fld)
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ (Base‘ℝ_fld))
19	16, 18	srasca 20798	. . . . . . . . . 10 ⊢ (⊤ → (ℝ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℝ_fld)‘ℝ)))
20	19	mptru 1549	. . . . . . . . 9 ⊢ (ℝ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℝ_fld)‘ℝ))
21	15, 20	eqtr3i 2763	. . . . . . . 8 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℝ_fld)‘ℝ))
22	13, 21	pwsval 17432	. . . . . . 7 ⊢ ((((subringAlg ‘ℝ_fld)‘ℝ) ∈ V ∧ 𝐼 ∈ 𝑉) → ((ringLMod‘ℝ_fld) ↑_s 𝐼) = (ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})))
23	8, 22	mpan 689	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((ringLMod‘ℝ_fld) ↑_s 𝐼) = (ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})))
24	23	eqcomd 2739	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) = ((ringLMod‘ℝ_fld) ↑_s 𝐼))
25	2	fveq2d 6896	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26		rrxbase.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐻)
27		eqid 2733	. . . . . . 7 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
28	27, 5	tcphbas 24736	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
29	25, 26, 28	3eqtr4g 2798	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
30	24, 29	oveq12d 7427	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵) = (((ringLMod‘ℝ_fld) ↑_s 𝐼) ↾_s (Base‘(ℝ_fld freeLMod 𝐼))))
31	7, 30	eqtr4d 2776	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) = ((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵))
32	31	fveq2d 6896	. 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵)))
33	2, 32	eqtrd 2773	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 {csn 4629 × cxp 5675 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 Basecbs 17144 ↾_s cress 17173 Scalarcsca 17200 X_scprds 17391 ↑_s cpws 17392 Fieldcfield 20358 subringAlg csra 20781 ringLModcrglmod 20782 ℝ_fldcrefld 21157 freeLMod cfrlm 21301 toℂPreHilctcph 24684 ℝ^crrx 24900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-prds 17393 df-pws 17395 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-subg 19003 df-cmn 19650 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-subrg 20317 df-drng 20359 df-field 20360 df-sra 20785 df-rgmod 20786 df-cnfld 20945 df-refld 21158 df-dsmm 21287 df-frlm 21302 df-tng 24093 df-tcph 24686 df-rrx 24902

This theorem is referenced by: rrxip 24907 rrxsca 24913

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