Theorem rrhval 30040

Description: Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.)

Hypotheses

Ref	Expression
rrhval.1	|- J = ( topGen ` ran (,) )
rrhval.2	|- K = ( TopOpen ` R )

Assertion

Ref	Expression
rrhval	|- ( R e. V -> (RRHom ` R ) = ( ( JCnExt K ) ` (QQHom ` R ) ) )

Proof of Theorem rrhval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( R e. V -> R e. _V )
2		rrhval.1	. . . . . . 7 |- J = ( topGen ` ran (,) )
3	2	eqcomi 2631	. . . . . 6 |- ( topGen ` ran (,) ) = J
4	3	a1i 11	. . . . 5 |- ( r = R -> ( topGen ` ran (,) ) = J )
5		fveq2 6191	. . . . . 6 |- ( r = R -> ( TopOpen ` r ) = ( TopOpen ` R ) )
6		rrhval.2	. . . . . 6 |- K = ( TopOpen ` R )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( r = R -> ( TopOpen ` r ) = K )
8	4, 7	oveq12d 6668	. . . 4 |- ( r = R -> ( ( topGen ` ran (,) )CnExt ( TopOpen ` r ) ) = ( JCnExt K ) )
9		fveq2 6191	. . . 4 |- ( r = R -> (QQHom ` r ) = (QQHom ` R ) )
10	8, 9	fveq12d 6197	. . 3 |- ( r = R -> ( ( ( topGen ` ran (,) )CnExt ( TopOpen ` r ) ) ` (QQHom ` r ) ) = ( ( JCnExt K ) ` (QQHom ` R ) ) )
11		df-rrh 30039	. . 3 |- RRHom = ( r e. _V |-> ( ( ( topGen ` ran (,) )CnExt ( TopOpen ` r ) ) ` (QQHom ` r ) ) )
12		fvex 6201	. . 3 |- ( ( JCnExt K ) ` (QQHom ` R ) ) e. _V
13	10, 11, 12	fvmpt 6282	. 2 |- ( R e. _V -> (RRHom ` R ) = ( ( JCnExt K ) ` (QQHom ` R ) ) )
14	1, 13	syl 17	1 |- ( R e. V -> (RRHom ` R ) = ( ( JCnExt K ) ` (QQHom ` R ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ran crn 5115   ` cfv 5888  (class class class)co 6650   (,)cioo 12175   TopOpenctopn 16082   topGenctg 16098  CnExtccnext 21863  QQHomcqqh 30016  RRHomcrrh 30037

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-rrh 30039

This theorem is referenced by:  rrhcn  30041  rrhqima  30058  rrhre  30065