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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 24579 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | refld 20852 | . . . . 5 ⊢ ℝfld ∈ Field | |
4 | eqid 2733 | . . . . . 6 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
5 | eqid 2733 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
6 | 4, 5 | frlmpws 20985 | . . . . 5 ⊢ ((ℝfld ∈ Field ∧ 𝐼 ∈ 𝑉) → (ℝfld freeLMod 𝐼) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
7 | 3, 6 | mpan 686 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
8 | fvex 6805 | . . . . . . 7 ⊢ ((subringAlg ‘ℝfld)‘ℝ) ∈ V | |
9 | rlmval 20489 | . . . . . . . . . 10 ⊢ (ringLMod‘ℝfld) = ((subringAlg ‘ℝfld)‘(Base‘ℝfld)) | |
10 | rebase 20839 | . . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝfld) | |
11 | 10 | fveq2i 6795 | . . . . . . . . . 10 ⊢ ((subringAlg ‘ℝfld)‘ℝ) = ((subringAlg ‘ℝfld)‘(Base‘ℝfld)) |
12 | 9, 11 | eqtr4i 2764 | . . . . . . . . 9 ⊢ (ringLMod‘ℝfld) = ((subringAlg ‘ℝfld)‘ℝ) |
13 | 12 | oveq1i 7305 | . . . . . . . 8 ⊢ ((ringLMod‘ℝfld) ↑s 𝐼) = (((subringAlg ‘ℝfld)‘ℝ) ↑s 𝐼) |
14 | 10 | ressid 16982 | . . . . . . . . . 10 ⊢ (ℝfld ∈ Field → (ℝfld ↾s ℝ) = ℝfld) |
15 | 3, 14 | ax-mp 5 | . . . . . . . . 9 ⊢ (ℝfld ↾s ℝ) = ℝfld |
16 | eqidd 2734 | . . . . . . . . . . 11 ⊢ (⊤ → ((subringAlg ‘ℝfld)‘ℝ) = ((subringAlg ‘ℝfld)‘ℝ)) | |
17 | 10 | eqimssi 3981 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ (Base‘ℝfld) |
18 | 17 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ (Base‘ℝfld)) |
19 | 16, 18 | srasca 20475 | . . . . . . . . . 10 ⊢ (⊤ → (ℝfld ↾s ℝ) = (Scalar‘((subringAlg ‘ℝfld)‘ℝ))) |
20 | 19 | mptru 1544 | . . . . . . . . 9 ⊢ (ℝfld ↾s ℝ) = (Scalar‘((subringAlg ‘ℝfld)‘ℝ)) |
21 | 15, 20 | eqtr3i 2763 | . . . . . . . 8 ⊢ ℝfld = (Scalar‘((subringAlg ‘ℝfld)‘ℝ)) |
22 | 13, 21 | pwsval 17225 | . . . . . . 7 ⊢ ((((subringAlg ‘ℝfld)‘ℝ) ∈ V ∧ 𝐼 ∈ 𝑉) → ((ringLMod‘ℝfld) ↑s 𝐼) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) |
23 | 8, 22 | mpan 686 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((ringLMod‘ℝfld) ↑s 𝐼) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) |
24 | 23 | eqcomd 2739 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) = ((ringLMod‘ℝfld) ↑s 𝐼)) |
25 | 2 | fveq2d 6796 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
26 | rrxbase.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐻) | |
27 | eqid 2733 | . . . . . . 7 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
28 | 27, 5 | tcphbas 24411 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
29 | 25, 26, 28 | 3eqtr4g 2798 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝfld freeLMod 𝐼))) |
30 | 24, 29 | oveq12d 7313 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵) = (((ringLMod‘ℝfld) ↑s 𝐼) ↾s (Base‘(ℝfld freeLMod 𝐼)))) |
31 | 7, 30 | eqtr4d 2776 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵)) |
32 | 31 | fveq2d 6796 | . 2 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
33 | 2, 32 | eqtrd 2773 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊤wtru 1538 ∈ wcel 2101 Vcvv 3434 ⊆ wss 3889 {csn 4564 × cxp 5589 ‘cfv 6447 (class class class)co 7295 ℝcr 10898 Basecbs 16940 ↾s cress 16969 Scalarcsca 16993 Xscprds 17184 ↑s cpws 17185 Fieldcfield 20020 subringAlg csra 20458 ringLModcrglmod 20459 ℝfldcrefld 20837 freeLMod cfrlm 20981 toℂPreHilctcph 24359 ℝ^crrx 24575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 ax-addf 10978 ax-mulf 10979 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-tpos 8062 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-rp 12759 df-fz 13268 df-seq 13750 df-exp 13811 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-hom 17014 df-cco 17015 df-0g 17180 df-prds 17186 df-pws 17188 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-grp 18608 df-minusg 18609 df-subg 18780 df-cmn 19416 df-mgp 19749 df-ur 19766 df-ring 19813 df-cring 19814 df-oppr 19890 df-dvdsr 19911 df-unit 19912 df-invr 19942 df-dvr 19953 df-drng 20021 df-field 20022 df-subrg 20050 df-sra 20462 df-rgmod 20463 df-cnfld 20626 df-refld 20838 df-dsmm 20967 df-frlm 20982 df-tng 23768 df-tcph 24361 df-rrx 24577 |
This theorem is referenced by: rrxip 24582 rrxsca 24588 |
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