![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rrxprds | Structured version Visualization version GIF version |
Description: Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
rrxval.r | ⊢ 𝐻 = (ℝ^‘𝐼) |
rrxbase.b | ⊢ 𝐵 = (Base‘𝐻) |
Ref | Expression |
---|---|
rrxprds | ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxval.r | . . 3 ⊢ 𝐻 = (ℝ^‘𝐼) | |
2 | 1 | rrxval 23510 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | refld 20285 | . . . . 5 ⊢ ℝfld ∈ Field | |
4 | eqid 2798 | . . . . . 6 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
5 | eqid 2798 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
6 | 4, 5 | frlmpws 20416 | . . . . 5 ⊢ ((ℝfld ∈ Field ∧ 𝐼 ∈ 𝑉) → (ℝfld freeLMod 𝐼) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
7 | 3, 6 | mpan 682 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
8 | fvex 6423 | . . . . . . 7 ⊢ ((subringAlg ‘ℝfld)‘ℝ) ∈ V | |
9 | rlmval 19511 | . . . . . . . . . 10 ⊢ (ringLMod‘ℝfld) = ((subringAlg ‘ℝfld)‘(Base‘ℝfld)) | |
10 | rebase 20272 | . . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝfld) | |
11 | 10 | fveq2i 6413 | . . . . . . . . . 10 ⊢ ((subringAlg ‘ℝfld)‘ℝ) = ((subringAlg ‘ℝfld)‘(Base‘ℝfld)) |
12 | 9, 11 | eqtr4i 2823 | . . . . . . . . 9 ⊢ (ringLMod‘ℝfld) = ((subringAlg ‘ℝfld)‘ℝ) |
13 | 12 | oveq1i 6887 | . . . . . . . 8 ⊢ ((ringLMod‘ℝfld) ↑s 𝐼) = (((subringAlg ‘ℝfld)‘ℝ) ↑s 𝐼) |
14 | 10 | ressid 16257 | . . . . . . . . . 10 ⊢ (ℝfld ∈ Field → (ℝfld ↾s ℝ) = ℝfld) |
15 | 3, 14 | ax-mp 5 | . . . . . . . . 9 ⊢ (ℝfld ↾s ℝ) = ℝfld |
16 | eqidd 2799 | . . . . . . . . . . 11 ⊢ (⊤ → ((subringAlg ‘ℝfld)‘ℝ) = ((subringAlg ‘ℝfld)‘ℝ)) | |
17 | 10 | eqimssi 3854 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ (Base‘ℝfld) |
18 | 17 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ (Base‘ℝfld)) |
19 | 16, 18 | srasca 19501 | . . . . . . . . . 10 ⊢ (⊤ → (ℝfld ↾s ℝ) = (Scalar‘((subringAlg ‘ℝfld)‘ℝ))) |
20 | 19 | mptru 1661 | . . . . . . . . 9 ⊢ (ℝfld ↾s ℝ) = (Scalar‘((subringAlg ‘ℝfld)‘ℝ)) |
21 | 15, 20 | eqtr3i 2822 | . . . . . . . 8 ⊢ ℝfld = (Scalar‘((subringAlg ‘ℝfld)‘ℝ)) |
22 | 13, 21 | pwsval 16458 | . . . . . . 7 ⊢ ((((subringAlg ‘ℝfld)‘ℝ) ∈ V ∧ 𝐼 ∈ 𝑉) → ((ringLMod‘ℝfld) ↑s 𝐼) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) |
23 | 8, 22 | mpan 682 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((ringLMod‘ℝfld) ↑s 𝐼) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) |
24 | 23 | eqcomd 2804 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) = ((ringLMod‘ℝfld) ↑s 𝐼)) |
25 | 2 | fveq2d 6414 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
26 | rrxbase.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐻) | |
27 | eqid 2798 | . . . . . . 7 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
28 | 27, 5 | tcphbas 23342 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
29 | 25, 26, 28 | 3eqtr4g 2857 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
30 | 24, 29 | oveq12d 6895 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
31 | 7, 30 | eqtr4d 2835 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵)) |
32 | 31 | fveq2d 6414 | . 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
33 | 2, 32 | eqtrd 2832 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 Vcvv 3384 ⊆ wss 3768 {csn 4367 × cxp 5309 ‘cfv 6100 (class class class)co 6877 ℝcr 10222 Basecbs 16181 ↾s cress 16182 Scalarcsca 16267 Xscprds 16418 ↑s cpws 16419 Fieldcfield 19063 subringAlg csra 19488 ringLModcrglmod 19489 ℝfldcrefld 20270 freeLMod cfrlm 20412 toℂPreHilctcph 23291 ℝ^crrx 23506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 ax-addf 10302 ax-mulf 10303 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-tpos 7589 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-map 8096 df-ixp 8148 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-sup 8589 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-rp 12072 df-fz 12578 df-seq 13053 df-exp 13112 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-0g 16414 df-prds 16420 df-pws 16422 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 df-subg 17901 df-cmn 18507 df-mgp 18803 df-ur 18815 df-ring 18862 df-cring 18863 df-oppr 18936 df-dvdsr 18954 df-unit 18955 df-invr 18985 df-dvr 18996 df-drng 19064 df-field 19065 df-subrg 19093 df-sra 19492 df-rgmod 19493 df-cnfld 20066 df-refld 20271 df-dsmm 20398 df-frlm 20413 df-tng 22714 df-tcph 23293 df-rrx 23508 |
This theorem is referenced by: rrxip 23513 |
Copyright terms: Public domain | W3C validator |