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Theorem frec2uz0d 9481
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 5210 . 2  |-  ( G `
 (/) )  =  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C ) `  (/) )
3 frec2uz.1 . . 3  |-  ( ph  ->  C  e.  ZZ )
4 frec0g 6046 . . 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 2126 1  |-  ( ph  ->  ( G `  (/) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   (/)c0 3258    |-> cmpt 3847   ` cfv 4932  (class class class)co 5543  freccfrec 6039   1c1 7044    + caddc 7046   ZZcz 8432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-recs 5954  df-frec 6040
This theorem is referenced by:  frec2uzuzd  9484  frec2uzrand  9487  frec2uzrdg  9491  frecuzrdgg  9498  frecfzennn  9508  omgadd  9826
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