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Theorem xrhval 27660
 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 RRHom
xrhval.l
xrhval.u
Assertion
Ref Expression
xrhval RR*Hom RRHom
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem xrhval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3122 . 2
2 eqidd 2468 . . . 4
3 fveq2 5866 . . . . . 6 RRHom RRHom
4 eqidd 2468 . . . . . 6
53, 4fveq12d 5872 . . . . 5 RRHom RRHom
6 fveq2 5866 . . . . . . . 8
7 xrhval.u . . . . . . . 8
86, 7syl6eqr 2526 . . . . . . 7
93imaeq1d 5336 . . . . . . . 8 RRHom RRHom
10 xrhval.b . . . . . . . 8 RRHom
119, 10syl6eqr 2526 . . . . . . 7 RRHom
128, 11fveq12d 5872 . . . . . 6 RRHom
13 fveq2 5866 . . . . . . . 8
14 xrhval.l . . . . . . . 8
1513, 14syl6eqr 2526 . . . . . . 7
1615, 11fveq12d 5872 . . . . . 6 RRHom
1712, 16ifeq12d 3959 . . . . 5 RRHom RRHom
185, 17ifeq12d 3959 . . . 4 RRHom RRHom RRHom RRHom
192, 18mpteq12dv 4525 . . 3 RRHom RRHom RRHom RRHom
20 df-xrh 27659 . . 3 RR*Hom RRHom RRHom RRHom
21 xrex 11217 . . . 4
2221mptex 6131 . . 3 RRHom
2319, 20, 22fvmpt 5950 . 2 RR*Hom RRHom
241, 23syl 16 1 RR*Hom RRHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1379   wcel 1767  cvv 3113  cif 3939   cmpt 4505  cima 5002  cfv 5588  cr 9491   cpnf 9625  cxr 9627  club 15429  cglb 15430  RRHomcrrh 27638  RR*Homcxrh 27658 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-xr 9632  df-xrh 27659 This theorem is referenced by: (None)
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