rrxprds - Metamath Proof Explorer

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

Theorem rrxprds 25267

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 25265	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		refld 21507	. . . . 5 ⊢ ℝ_fld ∈ Field
4		eqid 2726	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5		eqid 2726	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6	4, 5	frlmpws 21640	. . . . 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 687	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) = (((ringLMod‘ℝ_fld) ↑_s 𝐼) ↾_s (Base‘(ℝ_fld freeLMod 𝐼))))
8		fvex 6897	. . . . . . 7 ⊢ ((subringAlg ‘ℝ_fld)‘ℝ) ∈ V
9		rlmval 21044	. . . . . . . . . 10 ⊢ (ringLMod‘ℝ_fld) = ((subringAlg ‘ℝ_fld)‘(Base‘ℝ_fld))
10		rebase 21494	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
11	10	fveq2i 6887	. . . . . . . . . 10 ⊢ ((subringAlg ‘ℝ_fld)‘ℝ) = ((subringAlg ‘ℝ_fld)‘(Base‘ℝ_fld))
12	9, 11	eqtr4i 2757	. . . . . . . . 9 ⊢ (ringLMod‘ℝ_fld) = ((subringAlg ‘ℝ_fld)‘ℝ)
13	12	oveq1i 7414	. . . . . . . 8 ⊢ ((ringLMod‘ℝ_fld) ↑_s 𝐼) = (((subringAlg ‘ℝ_fld)‘ℝ) ↑_s 𝐼)
14	10	ressid 17195	. . . . . . . . . 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 2727	. . . . . . . . . . 11 ⊢ (⊤ → ((subringAlg ‘ℝ_fld)‘ℝ) = ((subringAlg ‘ℝ_fld)‘ℝ))
17	10	eqimssi 4037	. . . . . . . . . . . 12 ⊢ ℝ ⊆ (Base‘ℝ_fld)
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ (Base‘ℝ_fld))
19	16, 18	srasca 21029	. . . . . . . . . 10 ⊢ (⊤ → (ℝ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℝ_fld)‘ℝ)))
20	19	mptru 1540	. . . . . . . . 9 ⊢ (ℝ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℝ_fld)‘ℝ))
21	15, 20	eqtr3i 2756	. . . . . . . 8 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℝ_fld)‘ℝ))
22	13, 21	pwsval 17438	. . . . . . 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 687	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((ringLMod‘ℝ_fld) ↑_s 𝐼) = (ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})))
24	23	eqcomd 2732	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) = ((ringLMod‘ℝ_fld) ↑_s 𝐼))
25	2	fveq2d 6888	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26		rrxbase.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐻)
27		eqid 2726	. . . . . . 7 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
28	27, 5	tcphbas 25097	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
29	25, 26, 28	3eqtr4g 2791	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
30	24, 29	oveq12d 7422	. . . 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 2769	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) = ((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵))
32	31	fveq2d 6888	. 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘((ℝ_fldX_s(𝐼 × {((subringAlg ‘ℝ_fld)‘ℝ)})) ↾_s 𝐵)))
33	2, 32	eqtrd 2766	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 1533 ⊤wtru 1534 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 {csn 4623 × cxp 5667 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 Basecbs 17150 ↾_s cress 17179 Scalarcsca 17206 X_scprds 17397 ↑_s cpws 17398 Fieldcfield 20585 subringAlg csra 21016 ringLModcrglmod 21017 ℝ_fldcrefld 21492 freeLMod cfrlm 21636 toℂPreHilctcph 25045 ℝ^crrx 25261

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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188

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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-0g 17393 df-prds 17399 df-pws 17401 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-subg 19047 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-dvr 20300 df-subrng 20443 df-subrg 20468 df-drng 20586 df-field 20587 df-sra 21018 df-rgmod 21019 df-cnfld 21236 df-refld 21493 df-dsmm 21622 df-frlm 21637 df-tng 24443 df-tcph 25047 df-rrx 25263

This theorem is referenced by: rrxip 25268 rrxsca 25274

W3C validator