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Theorem frec2uz0d 10179
Description: The mapping  G is a one-to-one mapping from  om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number  C (normally 0 for the upper integers  NN0 or 1 for the upper integers  NN), 1 maps to  C + 1, etc. This theorem shows the value of  G at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
Hypotheses
Ref Expression
frec2uz.1  |-  ( ph  ->  C  e.  ZZ )
frec2uz.2  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
Assertion
Ref Expression
frec2uz0d  |-  ( ph  ->  ( G `  (/) )  =  C )
Distinct variable group:    x, C
Allowed substitution hints:    ph( x)    G( x)

Proof of Theorem frec2uz0d
StepHypRef Expression
1 frec2uz.2 . . 3  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
21fveq1i 5422 . 2  |-  ( G `
 (/) )  =  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C ) `  (/) )
3 frec2uz.1 . . 3  |-  ( ph  ->  C  e.  ZZ )
4 frec0g 6294 . . 3  |-  ( C  e.  ZZ  ->  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C ) `  (/) )  =  C )
53, 4syl 14 . 2  |-  ( ph  ->  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  C ) `  (/) )  =  C )
62, 5syl5eq 2184 1  |-  ( ph  ->  ( G `  (/) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   (/)c0 3363    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774  freccfrec 6287   1c1 7628    + caddc 7630   ZZcz 9061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-recs 6202  df-frec 6288
This theorem is referenced by:  frec2uzuzd  10182  frec2uzrand  10185  frec2uzrdg  10189  frecuzrdgg  10196  frecfzennn  10206  0tonninf  10219  1tonninf  10220  omgadd  10555  ennnfonelem1  11926  ennnfonelemhf1o  11932  isomninnlem  13278
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