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Theorem fztpval 9985
Description: Two ways of defining the first three values of a sequence on 
NN. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
fztpval  |-  ( A. x  e.  ( 1 ... 3 ) ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( ( F `  1 )  =  A  /\  ( F `  2 )  =  B  /\  ( F `  3 )  =  C ) )
Distinct variable groups:    x, A    x, B    x, C    x, F

Proof of Theorem fztpval
StepHypRef Expression
1 1z 9193 . . . . 5  |-  1  e.  ZZ
2 fztp 9980 . . . . 5  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
31, 2ax-mp 5 . . . 4  |-  ( 1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
4 df-3 8893 . . . . . 6  |-  3  =  ( 2  +  1 )
5 2cn 8904 . . . . . . 7  |-  2  e.  CC
6 ax-1cn 7825 . . . . . . 7  |-  1  e.  CC
75, 6addcomi 8019 . . . . . 6  |-  ( 2  +  1 )  =  ( 1  +  2 )
84, 7eqtri 2178 . . . . 5  |-  3  =  ( 1  +  2 )
98oveq2i 5835 . . . 4  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
10 tpeq3 3647 . . . . . 6  |-  ( 3  =  ( 1  +  2 )  ->  { 1 ,  2 ,  3 }  =  { 1 ,  2 ,  ( 1  +  2 ) } )
118, 10ax-mp 5 . . . . 5  |-  { 1 ,  2 ,  3 }  =  { 1 ,  2 ,  ( 1  +  2 ) }
12 df-2 8892 . . . . . 6  |-  2  =  ( 1  +  1 )
13 tpeq2 3646 . . . . . 6  |-  ( 2  =  ( 1  +  1 )  ->  { 1 ,  2 ,  ( 1  +  2 ) }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
1412, 13ax-mp 5 . . . . 5  |-  { 1 ,  2 ,  ( 1  +  2 ) }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
1511, 14eqtri 2178 . . . 4  |-  { 1 ,  2 ,  3 }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
163, 9, 153eqtr4i 2188 . . 3  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
1716raleqi 2656 . 2  |-  ( A. x  e.  ( 1 ... 3 ) ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  A. x  e.  { 1 ,  2 ,  3 }  ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C
) ) )
18 1ex 7873 . . 3  |-  1  e.  _V
19 2ex 8905 . . 3  |-  2  e.  _V
20 3ex 8909 . . 3  |-  3  e.  _V
21 fveq2 5468 . . . 4  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
22 iftrue 3510 . . . 4  |-  ( x  =  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  A )
2321, 22eqeq12d 2172 . . 3  |-  ( x  =  1  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
1 )  =  A ) )
24 fveq2 5468 . . . 4  |-  ( x  =  2  ->  ( F `  x )  =  ( F ` 
2 ) )
25 1re 7877 . . . . . . . 8  |-  1  e.  RR
26 1lt2 9002 . . . . . . . 8  |-  1  <  2
2725, 26gtneii 7972 . . . . . . 7  |-  2  =/=  1
28 neeq1 2340 . . . . . . 7  |-  ( x  =  2  ->  (
x  =/=  1  <->  2  =/=  1 ) )
2927, 28mpbiri 167 . . . . . 6  |-  ( x  =  2  ->  x  =/=  1 )
30 ifnefalse 3516 . . . . . 6  |-  ( x  =/=  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  if ( x  =  2 ,  B ,  C ) )
3129, 30syl 14 . . . . 5  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  if ( x  =  2 ,  B ,  C ) )
32 iftrue 3510 . . . . 5  |-  ( x  =  2  ->  if ( x  =  2 ,  B ,  C )  =  B )
3331, 32eqtrd 2190 . . . 4  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  B )
3424, 33eqeq12d 2172 . . 3  |-  ( x  =  2  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
2 )  =  B ) )
35 fveq2 5468 . . . 4  |-  ( x  =  3  ->  ( F `  x )  =  ( F ` 
3 ) )
36 1lt3 9004 . . . . . . . 8  |-  1  <  3
3725, 36gtneii 7972 . . . . . . 7  |-  3  =/=  1
38 neeq1 2340 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  1  <->  3  =/=  1 ) )
3937, 38mpbiri 167 . . . . . 6  |-  ( x  =  3  ->  x  =/=  1 )
4039, 30syl 14 . . . . 5  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  if ( x  =  2 ,  B ,  C ) )
41 2re 8903 . . . . . . . 8  |-  2  e.  RR
42 2lt3 9003 . . . . . . . 8  |-  2  <  3
4341, 42gtneii 7972 . . . . . . 7  |-  3  =/=  2
44 neeq1 2340 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  2  <->  3  =/=  2 ) )
4543, 44mpbiri 167 . . . . . 6  |-  ( x  =  3  ->  x  =/=  2 )
46 ifnefalse 3516 . . . . . 6  |-  ( x  =/=  2  ->  if ( x  =  2 ,  B ,  C )  =  C )
4745, 46syl 14 . . . . 5  |-  ( x  =  3  ->  if ( x  =  2 ,  B ,  C )  =  C )
4840, 47eqtrd 2190 . . . 4  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  C )
4935, 48eqeq12d 2172 . . 3  |-  ( x  =  3  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
3 )  =  C ) )
5018, 19, 20, 23, 34, 49raltp 3616 . 2  |-  ( A. x  e.  { 1 ,  2 ,  3 }  ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( ( F `
 1 )  =  A  /\  ( F `
 2 )  =  B  /\  ( F `
 3 )  =  C ) )
5117, 50bitri 183 1  |-  ( A. x  e.  ( 1 ... 3 ) ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( ( F `  1 )  =  A  /\  ( F `  2 )  =  B  /\  ( F `  3 )  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128    =/= wne 2327   A.wral 2435   ifcif 3505   {ctp 3562   ` cfv 5170  (class class class)co 5824   1c1 7733    + caddc 7735   2c2 8884   3c3 8885   ZZcz 9167   ...cfz 9912
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-tp 3568  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-2 8892  df-3 8893  df-n0 9091  df-z 9168  df-uz 9440  df-fz 9913
This theorem is referenced by: (None)
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