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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 3480 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | oveq2 7427 | . . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼)) | |
4 | 3 | fveq2d 6900 | . . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
5 | df-rrx 25357 | . . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) | |
6 | fvex 6909 | . . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V | |
7 | 4, 5, 6 | fvmpt 7004 | . . 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 | eqtrid 2777 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ‘cfv 6549 (class class class)co 7419 ℝfldcrefld 21553 freeLMod cfrlm 21697 toℂPreHilctcph 25139 ℝ^crrx 25355 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-rrx 25357 |
This theorem is referenced by: rrxbase 25360 rrxprds 25361 rrxnm 25363 rrxcph 25364 rrxds 25365 rrxvsca 25366 rrxplusgvscavalb 25367 rrx0 25369 rrxdim 33443 rrxtopn 45810 opnvonmbllem2 46159 |
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