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Theorem frec2uz0d 10401
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 5518 . 2  |-  ( G `
 (/) )  =  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C ) `  (/) )
3 frec2uz.1 . . 3  |-  ( ph  ->  C  e.  ZZ )
4 frec0g 6400 . . 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, 5eqtrid 2222 1  |-  ( ph  ->  ( G `  (/) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   (/)c0 3424    |-> cmpt 4066   ` cfv 5218  (class class class)co 5877  freccfrec 6393   1c1 7814    + caddc 7816   ZZcz 9255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-recs 6308  df-frec 6394
This theorem is referenced by:  frec2uzuzd  10404  frec2uzrand  10407  frec2uzrdg  10411  frecuzrdgg  10418  frecfzennn  10428  0tonninf  10441  1tonninf  10442  omgadd  10784  ennnfonelem1  12410  ennnfonelemhf1o  12416  012of  14784  2o01f  14785  isomninnlem  14817  iswomninnlem  14836  ismkvnnlem  14839
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