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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 3501 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
| 3 | oveq2 7439 | . . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼)) | |
| 4 | 3 | fveq2d 6910 | . . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
| 5 | df-rrx 25419 | . . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) | |
| 6 | fvex 6919 | . . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V | |
| 7 | 4, 5, 6 | fvmpt 7016 | . . 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 2789 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 ℝfldcrefld 21622 freeLMod cfrlm 21766 toℂPreHilctcph 25201 ℝ^crrx 25417 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-rrx 25419 |
| This theorem is referenced by: rrxbase 25422 rrxprds 25423 rrxnm 25425 rrxcph 25426 rrxds 25427 rrxvsca 25428 rrxplusgvscavalb 25429 rrx0 25431 rrxdim 33665 rrxtopn 46299 opnvonmbllem2 46648 |
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