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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 23991 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | refld 20308 | . . . . 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 20439 | . . . . 5 ⊢ ((ℝfld ∈ Field ∧ 𝐼 ∈ 𝑉) → (ℝfld freeLMod 𝐼) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
7 | 3, 6 | mpan 689 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
8 | fvex 6658 | . . . . . . 7 ⊢ ((subringAlg ‘ℝfld)‘ℝ) ∈ V | |
9 | rlmval 19956 | . . . . . . . . . 10 ⊢ (ringLMod‘ℝfld) = ((subringAlg ‘ℝfld)‘(Base‘ℝfld)) | |
10 | rebase 20295 | . . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝfld) | |
11 | 10 | fveq2i 6648 | . . . . . . . . . 10 ⊢ ((subringAlg ‘ℝfld)‘ℝ) = ((subringAlg ‘ℝfld)‘(Base‘ℝfld)) |
12 | 9, 11 | eqtr4i 2824 | . . . . . . . . 9 ⊢ (ringLMod‘ℝfld) = ((subringAlg ‘ℝfld)‘ℝ) |
13 | 12 | oveq1i 7145 | . . . . . . . 8 ⊢ ((ringLMod‘ℝfld) ↑s 𝐼) = (((subringAlg ‘ℝfld)‘ℝ) ↑s 𝐼) |
14 | 10 | ressid 16551 | . . . . . . . . . 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 3973 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ (Base‘ℝfld) |
18 | 17 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ (Base‘ℝfld)) |
19 | 16, 18 | srasca 19946 | . . . . . . . . . 10 ⊢ (⊤ → (ℝfld ↾s ℝ) = (Scalar‘((subringAlg ‘ℝfld)‘ℝ))) |
20 | 19 | mptru 1545 | . . . . . . . . 9 ⊢ (ℝfld ↾s ℝ) = (Scalar‘((subringAlg ‘ℝfld)‘ℝ)) |
21 | 15, 20 | eqtr3i 2823 | . . . . . . . 8 ⊢ ℝfld = (Scalar‘((subringAlg ‘ℝfld)‘ℝ)) |
22 | 13, 21 | pwsval 16751 | . . . . . . 7 ⊢ ((((subringAlg ‘ℝfld)‘ℝ) ∈ V ∧ 𝐼 ∈ 𝑉) → ((ringLMod‘ℝfld) ↑s 𝐼) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) |
23 | 8, 22 | mpan 689 | . . . . . 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 6649 | . . . . . 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 23823 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
29 | 25, 26, 28 | 3eqtr4g 2858 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
30 | 24, 29 | oveq12d 7153 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
31 | 7, 30 | eqtr4d 2836 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵)) |
32 | 31 | fveq2d 6649 | . 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
33 | 2, 32 | eqtrd 2833 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 Basecbs 16475 ↾s cress 16476 Scalarcsca 16560 Xscprds 16711 ↑s cpws 16712 Fieldcfield 19496 subringAlg csra 19933 ringLModcrglmod 19934 ℝfldcrefld 20293 freeLMod cfrlm 20435 toℂPreHilctcph 23772 ℝ^crrx 23987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-field 19498 df-subrg 19526 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-tng 23191 df-tcph 23774 df-rrx 23989 |
This theorem is referenced by: rrxip 23994 rrxsca 24000 |
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