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Theorem fztpval 10039
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 9238 . . . . 5  |-  1  e.  ZZ
2 fztp 10034 . . . . 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 8938 . . . . . 6  |-  3  =  ( 2  +  1 )
5 2cn 8949 . . . . . . 7  |-  2  e.  CC
6 ax-1cn 7867 . . . . . . 7  |-  1  e.  CC
75, 6addcomi 8063 . . . . . 6  |-  ( 2  +  1 )  =  ( 1  +  2 )
84, 7eqtri 2191 . . . . 5  |-  3  =  ( 1  +  2 )
98oveq2i 5864 . . . 4  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
10 tpeq3 3671 . . . . . 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 8937 . . . . . 6  |-  2  =  ( 1  +  1 )
13 tpeq2 3670 . . . . . 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 2191 . . . 4  |-  { 1 ,  2 ,  3 }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
163, 9, 153eqtr4i 2201 . . 3  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
1716raleqi 2669 . 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 7915 . . 3  |-  1  e.  _V
19 2ex 8950 . . 3  |-  2  e.  _V
20 3ex 8954 . . 3  |-  3  e.  _V
21 fveq2 5496 . . . 4  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
22 iftrue 3531 . . . 4  |-  ( x  =  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  A )
2321, 22eqeq12d 2185 . . 3  |-  ( x  =  1  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
1 )  =  A ) )
24 fveq2 5496 . . . 4  |-  ( x  =  2  ->  ( F `  x )  =  ( F ` 
2 ) )
25 1re 7919 . . . . . . . 8  |-  1  e.  RR
26 1lt2 9047 . . . . . . . 8  |-  1  <  2
2725, 26gtneii 8015 . . . . . . 7  |-  2  =/=  1
28 neeq1 2353 . . . . . . 7  |-  ( x  =  2  ->  (
x  =/=  1  <->  2  =/=  1 ) )
2927, 28mpbiri 167 . . . . . 6  |-  ( x  =  2  ->  x  =/=  1 )
30 ifnefalse 3537 . . . . . 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 3531 . . . . 5  |-  ( x  =  2  ->  if ( x  =  2 ,  B ,  C )  =  B )
3331, 32eqtrd 2203 . . . 4  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  B )
3424, 33eqeq12d 2185 . . 3  |-  ( x  =  2  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
2 )  =  B ) )
35 fveq2 5496 . . . 4  |-  ( x  =  3  ->  ( F `  x )  =  ( F ` 
3 ) )
36 1lt3 9049 . . . . . . . 8  |-  1  <  3
3725, 36gtneii 8015 . . . . . . 7  |-  3  =/=  1
38 neeq1 2353 . . . . . . 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 8948 . . . . . . . 8  |-  2  e.  RR
42 2lt3 9048 . . . . . . . 8  |-  2  <  3
4341, 42gtneii 8015 . . . . . . 7  |-  3  =/=  2
44 neeq1 2353 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  2  <->  3  =/=  2 ) )
4543, 44mpbiri 167 . . . . . 6  |-  ( x  =  3  ->  x  =/=  2 )
46 ifnefalse 3537 . . . . . 6  |-  ( x  =/=  2  ->  if ( x  =  2 ,  B ,  C )  =  C )
4745, 46syl 14 . . . . 5  |-  ( x  =  3  ->  if ( x  =  2 ,  B ,  C )  =  C )
4840, 47eqtrd 2203 . . . 4  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  C )
4935, 48eqeq12d 2185 . . 3  |-  ( x  =  3  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
3 )  =  C ) )
5018, 19, 20, 23, 34, 49raltp 3640 . 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 973    = wceq 1348    e. wcel 2141    =/= wne 2340   A.wral 2448   ifcif 3526   {ctp 3585   ` cfv 5198  (class class class)co 5853   1c1 7775    + caddc 7777   2c2 8929   3c3 8930   ZZcz 9212   ...cfz 9965
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-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  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-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-tp 3591  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-3 8938  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966
This theorem is referenced by: (None)
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