rrxval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rrxval		Structured version Visualization version GIF version

Theorem rrxval 25434

Description: Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)

**Hypothesis**
Ref	Expression
rrxval.r	⊢ 𝐻 = (ℝ^‘𝐼)

**Assertion**
Ref	Expression
rrxval	⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))

Proof of Theorem rrxval Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxval.r	. 2 ⊢ 𝐻 = (ℝ^‘𝐼)
2		elex 3498	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7438	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝ_fld freeLMod 𝑖) = (ℝ_fld freeLMod 𝐼))
4	3	fveq2d 6910	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝ_fld freeLMod 𝑖)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5		df-rrx 25432	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
6		fvex 6919	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 7015	. . 3 ⊢ (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
8	2, 7	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
9	1, 8	eqtrid 2786	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ‘cfv 6562 (class class class)co 7430 ℝ_fldcrefld 21639 freeLMod cfrlm 21783 toℂPreHilctcph 25214 ℝ^crrx 25430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-rrx 25432

This theorem is referenced by: rrxbase 25435 rrxprds 25436 rrxnm 25438 rrxcph 25439 rrxds 25440 rrxvsca 25441 rrxplusgvscavalb 25442 rrx0 25444 rrxdim 33641 rrxtopn 46239 opnvonmbllem2 46588

W3C validator