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Theorem fztpval 10845
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 10053 . . . . 5  |-  1  e.  ZZ
2 fztp 10841 . . . . 5  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
31, 2ax-mp 8 . . . 4  |-  ( 1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
4 df-3 9805 . . . . . 6  |-  3  =  ( 2  +  1 )
5 2cn 9816 . . . . . . 7  |-  2  e.  CC
6 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
75, 6addcomi 9003 . . . . . 6  |-  ( 2  +  1 )  =  ( 1  +  2 )
84, 7eqtri 2303 . . . . 5  |-  3  =  ( 1  +  2 )
98oveq2i 5869 . . . 4  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
10 tpeq3 3717 . . . . . 6  |-  ( 3  =  ( 1  +  2 )  ->  { 1 ,  2 ,  3 }  =  { 1 ,  2 ,  ( 1  +  2 ) } )
118, 10ax-mp 8 . . . . 5  |-  { 1 ,  2 ,  3 }  =  { 1 ,  2 ,  ( 1  +  2 ) }
12 df-2 9804 . . . . . 6  |-  2  =  ( 1  +  1 )
13 tpeq2 3716 . . . . . 6  |-  ( 2  =  ( 1  +  1 )  ->  { 1 ,  2 ,  ( 1  +  2 ) }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
1412, 13ax-mp 8 . . . . 5  |-  { 1 ,  2 ,  ( 1  +  2 ) }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
1511, 14eqtri 2303 . . . 4  |-  { 1 ,  2 ,  3 }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
163, 9, 153eqtr4i 2313 . . 3  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
1716raleqi 2740 . 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 8833 . . 3  |-  1  e.  _V
195elexi 2797 . . 3  |-  2  e.  _V
20 3re 9817 . . . 4  |-  3  e.  RR
2120elexi 2797 . . 3  |-  3  e.  _V
22 fveq2 5525 . . . 4  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
23 iftrue 3571 . . . 4  |-  ( x  =  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  A )
2422, 23eqeq12d 2297 . . 3  |-  ( x  =  1  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
1 )  =  A ) )
25 fveq2 5525 . . . 4  |-  ( x  =  2  ->  ( F `  x )  =  ( F ` 
2 ) )
26 1re 8837 . . . . . . . 8  |-  1  e.  RR
27 1lt2 9886 . . . . . . . 8  |-  1  <  2
2826, 27gtneii 8930 . . . . . . 7  |-  2  =/=  1
29 neeq1 2454 . . . . . . 7  |-  ( x  =  2  ->  (
x  =/=  1  <->  2  =/=  1 ) )
3028, 29mpbiri 224 . . . . . 6  |-  ( x  =  2  ->  x  =/=  1 )
31 ifnefalse 3573 . . . . . 6  |-  ( x  =/=  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  if ( x  =  2 ,  B ,  C ) )
3230, 31syl 15 . . . . 5  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  if ( x  =  2 ,  B ,  C ) )
33 iftrue 3571 . . . . 5  |-  ( x  =  2  ->  if ( x  =  2 ,  B ,  C )  =  B )
3432, 33eqtrd 2315 . . . 4  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  B )
3525, 34eqeq12d 2297 . . 3  |-  ( x  =  2  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
2 )  =  B ) )
36 fveq2 5525 . . . 4  |-  ( x  =  3  ->  ( F `  x )  =  ( F ` 
3 ) )
37 1lt3 9888 . . . . . . . 8  |-  1  <  3
3826, 37gtneii 8930 . . . . . . 7  |-  3  =/=  1
39 neeq1 2454 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  1  <->  3  =/=  1 ) )
4038, 39mpbiri 224 . . . . . 6  |-  ( x  =  3  ->  x  =/=  1 )
4140, 31syl 15 . . . . 5  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  if ( x  =  2 ,  B ,  C ) )
42 2re 9815 . . . . . . . 8  |-  2  e.  RR
43 2lt3 9887 . . . . . . . 8  |-  2  <  3
4442, 43gtneii 8930 . . . . . . 7  |-  3  =/=  2
45 neeq1 2454 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  2  <->  3  =/=  2 ) )
4644, 45mpbiri 224 . . . . . 6  |-  ( x  =  3  ->  x  =/=  2 )
47 ifnefalse 3573 . . . . . 6  |-  ( x  =/=  2  ->  if ( x  =  2 ,  B ,  C )  =  C )
4846, 47syl 15 . . . . 5  |-  ( x  =  3  ->  if ( x  =  2 ,  B ,  C )  =  C )
4941, 48eqtrd 2315 . . . 4  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  C )
5036, 49eqeq12d 2297 . . 3  |-  ( x  =  3  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
3 )  =  C ) )
5118, 19, 21, 24, 35, 50raltp 3688 . 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 ) )
5217, 51bitri 240 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ifcif 3565   {ctp 3642   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740   2c2 9795   3c3 9796   ZZcz 10024   ...cfz 10782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783
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