Theorem rrxval 25546

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‘(ℝfld 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 3476	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7418	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
4	3	fveq2d 6885	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5		df-rrx 25544	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6		fvex 6894	. . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6989	. . 3 ⊢ (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
8	2, 7	syl 18	. 2 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
9	1, 8	eqtrid 2810	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ‘cfv 6536  (class class class)co 7410  ℝ_fldcrefld 21754   freeLMod cfrlm 21896  toℂPreHilctcph 25326  ℝ^crrx 25542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-rrx 25544

This theorem is referenced by:  rrxbase  25547  rrxprds  25548  rrxnm  25550  rrxcph  25551  rrxds  25552  rrxvsca  25553  rrxplusgvscavalb  25554  rrx0  25556  rrxdim  34004  rrxtopn  46998  opnvonmbllem2  47347