Theorem frec2uz0d 10355

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

Step	Hyp	Ref	Expression
1		frec2uz.2	. . 3 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
2	1	fveq1i 5497	. 2 |- ( G ` (/) ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) )
3		frec2uz.1	. . 3 |- ( ph -> C e. ZZ )
4		frec0g 6376	. . 3 |- ( C e. ZZ -> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) ) = C )
5	3, 4	syl 14	. 2 |- ( ph -> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , C ) ` (/) ) = C )
6	2, 5	eqtrid 2215	1 |- ( ph -> ( G ` (/) ) = C )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   (/)c0 3414   |-> cmpt 4050   ` cfv 5198  (class class class)co 5853  freccfrec 6369   1c1 7775   + caddc 7777   ZZcz 9212

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-recs 6284  df-frec 6370

This theorem is referenced by:  frec2uzuzd  10358  frec2uzrand  10361  frec2uzrdg  10365  frecuzrdgg  10372  frecfzennn  10382  0tonninf  10395  1tonninf  10396  omgadd  10737  ennnfonelem1  12362  ennnfonelemhf1o  12368  012of  14028  2o01f  14029  isomninnlem  14062  iswomninnlem  14081  ismkvnnlem  14084