Theorem rrxval 23175

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 e. V -> H = (toCHil ` (RRfld 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 3212	. . 3 |- ( I e. V -> I e. _V )
3		oveq2 6658	. . . . 5 |- ( i = I -> (RRfld freeLMod i ) = (RRfld freeLMod I ) )
4	3	fveq2d 6195	. . . 4 |- ( i = I -> (toCHil ` (RRfld freeLMod i ) ) = (toCHil ` (RRfld freeLMod I ) ) )
5		df-rrx 23173	. . . 4 |- ℝ^ = ( i e. _V |-> (toCHil ` (RRfld freeLMod i ) ) )
6		fvex 6201	. . . 4 |- (toCHil ` (RRfld freeLMod I ) ) e. _V
7	4, 5, 6	fvmpt 6282	. . 3 |- ( I e. _V -> (ℝ^ ` I ) = (toCHil ` (RRfld freeLMod I ) ) )
8	2, 7	syl 17	. 2 |- ( I e. V -> (ℝ^ ` I ) = (toCHil ` (RRfld freeLMod I ) ) )
9	1, 8	syl5eq 2668	1 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650  RRfldcrefld 19950   freeLMod cfrlm 20090  toCHilctch 22967  ℝ^crrx 23171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-rrx 23173

This theorem is referenced by:  rrxbase  23176  rrxprds  23177  rrxnm  23179  rrxcph  23180  rrxds  23181  rrxtopn  40501  opnvonmbllem2  40847