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Theorem fztpval 10861
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 10069 . . . . 5  |-  1  e.  ZZ
2 fztp 10857 . . . . 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 9821 . . . . . 6  |-  3  =  ( 2  +  1 )
5 2cn 9832 . . . . . . 7  |-  2  e.  CC
6 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
75, 6addcomi 9019 . . . . . 6  |-  ( 2  +  1 )  =  ( 1  +  2 )
84, 7eqtri 2316 . . . . 5  |-  3  =  ( 1  +  2 )
98oveq2i 5885 . . . 4  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
10 tpeq3 3730 . . . . . 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 9820 . . . . . 6  |-  2  =  ( 1  +  1 )
13 tpeq2 3729 . . . . . 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 2316 . . . 4  |-  { 1 ,  2 ,  3 }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
163, 9, 153eqtr4i 2326 . . 3  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
1716raleqi 2753 . 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 8849 . . 3  |-  1  e.  _V
195elexi 2810 . . 3  |-  2  e.  _V
20 3re 9833 . . . 4  |-  3  e.  RR
2120elexi 2810 . . 3  |-  3  e.  _V
22 fveq2 5541 . . . 4  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
23 iftrue 3584 . . . 4  |-  ( x  =  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  A )
2422, 23eqeq12d 2310 . . 3  |-  ( x  =  1  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
1 )  =  A ) )
25 fveq2 5541 . . . 4  |-  ( x  =  2  ->  ( F `  x )  =  ( F ` 
2 ) )
26 1re 8853 . . . . . . . 8  |-  1  e.  RR
27 1lt2 9902 . . . . . . . 8  |-  1  <  2
2826, 27gtneii 8946 . . . . . . 7  |-  2  =/=  1
29 neeq1 2467 . . . . . . 7  |-  ( x  =  2  ->  (
x  =/=  1  <->  2  =/=  1 ) )
3028, 29mpbiri 224 . . . . . 6  |-  ( x  =  2  ->  x  =/=  1 )
31 ifnefalse 3586 . . . . . 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 3584 . . . . 5  |-  ( x  =  2  ->  if ( x  =  2 ,  B ,  C )  =  B )
3432, 33eqtrd 2328 . . . 4  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  B )
3525, 34eqeq12d 2310 . . 3  |-  ( x  =  2  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
2 )  =  B ) )
36 fveq2 5541 . . . 4  |-  ( x  =  3  ->  ( F `  x )  =  ( F ` 
3 ) )
37 1lt3 9904 . . . . . . . 8  |-  1  <  3
3826, 37gtneii 8946 . . . . . . 7  |-  3  =/=  1
39 neeq1 2467 . . . . . . 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 9831 . . . . . . . 8  |-  2  e.  RR
43 2lt3 9903 . . . . . . . 8  |-  2  <  3
4442, 43gtneii 8946 . . . . . . 7  |-  3  =/=  2
45 neeq1 2467 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  2  <->  3  =/=  2 ) )
4644, 45mpbiri 224 . . . . . 6  |-  ( x  =  3  ->  x  =/=  2 )
47 ifnefalse 3586 . . . . . 6  |-  ( x  =/=  2  ->  if ( x  =  2 ,  B ,  C )  =  C )
4846, 47syl 15 . . . . 5  |-  ( x  =  3  ->  if ( x  =  2 ,  B ,  C )  =  C )
4941, 48eqtrd 2328 . . . 4  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  C )
5036, 49eqeq12d 2310 . . 3  |-  ( x  =  3  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
3 )  =  C ) )
5118, 19, 21, 24, 35, 50raltp 3701 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   ifcif 3578   {ctp 3655   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756   2c2 9811   3c3 9812   ZZcz 10040   ...cfz 10798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799
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