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Theorem xrhval 30062
Description: The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
Hypotheses
Ref Expression
xrhval.b |- B = ( (RRHom ` R ) " RR )
xrhval.l |- L = ( glb ` R )
xrhval.u |- U = ( lub ` R )
Assertion
Ref Expression
xrhval |- ( R e. V -> (RR*Hom ` R ) = ( x e. RR* |-> if ( x e. RR , ( (RRHom ` R ) ` x ) , if ( x = +oo , ( U ` B ) , ( L ` B ) ) ) ) )
Distinct variable group:   x, R
Allowed substitution hints:   B( x)   U( x)   L( x)   V( x)
Proof of Theorem xrhval
Dummy variable r is distinct from all other variables.
StepHypRef Expression
1 elex 3212 . 2 |- ( R e. V -> R e. _V )
2 fveq2 6191 . . . . . 6 |- ( r = R -> (RRHom ` r ) = (RRHom ` R ) )
32fveq1d 6193 . . . . 5 |- ( r = R -> ( (RRHom ` r ) ` x ) = ( (RRHom ` R ) ` x ) )
4 fveq2 6191 . . . . . . . 8 |- ( r = R -> ( lub ` r ) = ( lub ` R ) )
5 xrhval.u . . . . . . . 8 |- U = ( lub ` R )
64, 5syl6eqr 2674 . . . . . . 7 |- ( r = R -> ( lub ` r ) = U )
72imaeq1d 5465 . . . . . . . 8 |- ( r = R -> ( (RRHom ` r ) " RR ) = ( (RRHom ` R ) " RR ) )
8 xrhval.b . . . . . . . 8 |- B = ( (RRHom ` R ) " RR )
97, 8syl6eqr 2674 . . . . . . 7 |- ( r = R -> ( (RRHom ` r ) " RR ) = B )
106, 9fveq12d 6197 . . . . . 6 |- ( r = R -> ( ( lub ` r ) ` ( (RRHom ` r ) " RR ) ) = ( U ` B ) )
11 fveq2 6191 . . . . . . . 8 |- ( r = R -> ( glb ` r ) = ( glb ` R ) )
12 xrhval.l . . . . . . . 8 |- L = ( glb ` R )
1311, 12syl6eqr 2674 . . . . . . 7 |- ( r = R -> ( glb ` r ) = L )
1413, 9fveq12d 6197 . . . . . 6 |- ( r = R -> ( ( glb ` r ) ` ( (RRHom ` r ) " RR ) ) = ( L ` B ) )
1510, 14ifeq12d 4106 . . . . 5 |- ( r = R -> if ( x = +oo , ( ( lub ` r ) ` ( (RRHom ` r ) " RR ) ) , ( ( glb ` r ) ` ( (RRHom ` r ) " RR ) ) ) = if ( x = +oo , ( U ` B ) , ( L ` B ) ) )
163, 15ifeq12d 4106 . . . 4 |- ( r = R -> if ( x e. RR , ( (RRHom ` r ) ` x ) , if ( x = +oo , ( ( lub ` r ) ` ( (RRHom ` r ) " RR ) ) , ( ( glb ` r ) ` ( (RRHom ` r ) " RR ) ) ) ) = if ( x e. RR , ( (RRHom ` R ) ` x ) , if ( x = +oo , ( U ` B ) , ( L ` B ) ) ) )
1716mpteq2dv 4745 . . 3 |- ( r = R -> ( x e. RR* |-> if ( x e. RR , ( (RRHom ` r ) ` x ) , if ( x = +oo , ( ( lub ` r ) ` ( (RRHom ` r ) " RR ) ) , ( ( glb ` r ) ` ( (RRHom ` r ) " RR ) ) ) ) ) = ( x e. RR* |-> if ( x e. RR , ( (RRHom ` R ) ` x ) , if ( x = +oo , ( U ` B ) , ( L ` B ) ) ) ) )
18 df-xrh 30061 . . 3 |- RR*Hom = ( r e. _V |-> ( x e. RR* |-> if ( x e. RR , ( (RRHom ` r ) ` x ) , if ( x = +oo , ( ( lub ` r ) ` ( (RRHom ` r ) " RR ) ) , ( ( glb ` r ) ` ( (RRHom ` r ) " RR ) ) ) ) ) )
19 xrex 11829 . . . 4 |- RR* e. _V
2019mptex 6486 . . 3 |- ( x e. RR* |-> if ( x e. RR , ( (RRHom ` R ) ` x ) , if ( x = +oo , ( U ` B ) , ( L ` B ) ) ) ) e. _V
2117, 18, 20fvmpt 6282 . 2 |- ( R e. _V -> (RR*Hom ` R ) = ( x e. RR* |-> if ( x e. RR , ( (RRHom ` R ) ` x ) , if ( x = +oo , ( U ` B ) , ( L ` B ) ) ) ) )
221, 21syl 17 1 |- ( R e. V -> (RR*Hom ` R ) = ( x e. RR* |-> if ( x e. RR , ( (RRHom ` R ) ` x ) , if ( x = +oo , ( U ` B ) , ( L ` B ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   "cima 5117   ` cfv 5888   RRcr 9935   +oocpnf 10071   RR*cxr 10073   lubclub 16942   glbcglb 16943  RRHomcrrh 30037  RR*Homcxrh 30060
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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993
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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-xr 10078  df-xrh 30061
This theorem is referenced by: (None)
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