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Mirrors > Home > MPE Home > Th. List > rrxval | Structured version Visualization version GIF version |
Description: Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
rrxval.r | ⊢ 𝐻 = (ℝ^‘𝐼) |
Ref | Expression |
---|---|
rrxval | ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxval.r | . 2 ⊢ 𝐻 = (ℝ^‘𝐼) | |
2 | elex 3510 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | oveq2 7153 | . . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼)) | |
4 | 3 | fveq2d 6667 | . . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | df-rrx 23915 | . . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) | |
6 | fvex 6676 | . . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V | |
7 | 4, 5, 6 | fvmpt 6761 | . . 3 ⊢ (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
9 | 1, 8 | syl5eq 2865 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ‘cfv 6348 (class class class)co 7145 ℝfldcrefld 20676 freeLMod cfrlm 20818 toℂPreHilctcph 23698 ℝ^crrx 23913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-rrx 23915 |
This theorem is referenced by: rrxbase 23918 rrxprds 23919 rrxnm 23921 rrxcph 23922 rrxds 23923 rrxvsca 23924 rrxplusgvscavalb 23925 rrx0 23927 rrxdim 30911 rrxtopn 42446 opnvonmbllem2 42792 |
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