Theorem rrxval 25717

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 3471	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7432	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
4	3	fveq2d 6894	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5		df-rrx 25715	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6		fvex 6903	. . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6998	. . 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 2807	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ‘cfv 6543  (class class class)co 7424  ℝ_fldcrefld 21919   freeLMod cfrlm 22061  toℂPreHilctcph 25497  ℝ^crrx 25713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-rrx 25715

This theorem is used by:  rrxbase  25718  rrxprds  25719  rrxnm  25721  rrxcph  25722  rrxds  25723  rrxvsca  25724  rrxplusgvscavalb  25725  rrx0  25727  rrxdim  34296  rrxtopn  47402  opnvonmbllem2  47751