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Theorem fztpval 9419
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 8701 . . . . 5  |-  1  e.  ZZ
2 fztp 9414 . . . . 5  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
31, 2ax-mp 7 . . . 4  |-  ( 1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
4 df-3 8409 . . . . . 6  |-  3  =  ( 2  +  1 )
5 2cn 8420 . . . . . . 7  |-  2  e.  CC
6 ax-1cn 7374 . . . . . . 7  |-  1  e.  CC
75, 6addcomi 7562 . . . . . 6  |-  ( 2  +  1 )  =  ( 1  +  2 )
84, 7eqtri 2105 . . . . 5  |-  3  =  ( 1  +  2 )
98oveq2i 5617 . . . 4  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
10 tpeq3 3512 . . . . . 6  |-  ( 3  =  ( 1  +  2 )  ->  { 1 ,  2 ,  3 }  =  { 1 ,  2 ,  ( 1  +  2 ) } )
118, 10ax-mp 7 . . . . 5  |-  { 1 ,  2 ,  3 }  =  { 1 ,  2 ,  ( 1  +  2 ) }
12 df-2 8408 . . . . . 6  |-  2  =  ( 1  +  1 )
13 tpeq2 3511 . . . . . 6  |-  ( 2  =  ( 1  +  1 )  ->  { 1 ,  2 ,  ( 1  +  2 ) }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
1412, 13ax-mp 7 . . . . 5  |-  { 1 ,  2 ,  ( 1  +  2 ) }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
1511, 14eqtri 2105 . . . 4  |-  { 1 ,  2 ,  3 }  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
163, 9, 153eqtr4i 2115 . . 3  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
1716raleqi 2562 . 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 7419 . . 3  |-  1  e.  _V
19 2ex 8421 . . 3  |-  2  e.  _V
20 3ex 8425 . . 3  |-  3  e.  _V
21 fveq2 5261 . . . 4  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
22 iftrue 3384 . . . 4  |-  ( x  =  1  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  A )
2321, 22eqeq12d 2099 . . 3  |-  ( x  =  1  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
1 )  =  A ) )
24 fveq2 5261 . . . 4  |-  ( x  =  2  ->  ( F `  x )  =  ( F ` 
2 ) )
25 1re 7423 . . . . . . . 8  |-  1  e.  RR
26 1lt2 8511 . . . . . . . 8  |-  1  <  2
2725, 26gtneii 7516 . . . . . . 7  |-  2  =/=  1
28 neeq1 2264 . . . . . . 7  |-  ( x  =  2  ->  (
x  =/=  1  <->  2  =/=  1 ) )
2927, 28mpbiri 166 . . . . . 6  |-  ( x  =  2  ->  x  =/=  1 )
30 ifnefalse 3390 . . . . . 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 3384 . . . . 5  |-  ( x  =  2  ->  if ( x  =  2 ,  B ,  C )  =  B )
3331, 32eqtrd 2117 . . . 4  |-  ( x  =  2  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  B )
3424, 33eqeq12d 2099 . . 3  |-  ( x  =  2  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
2 )  =  B ) )
35 fveq2 5261 . . . 4  |-  ( x  =  3  ->  ( F `  x )  =  ( F ` 
3 ) )
36 1lt3 8513 . . . . . . . 8  |-  1  <  3
3725, 36gtneii 7516 . . . . . . 7  |-  3  =/=  1
38 neeq1 2264 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  1  <->  3  =/=  1 ) )
3937, 38mpbiri 166 . . . . . 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 8419 . . . . . . . 8  |-  2  e.  RR
42 2lt3 8512 . . . . . . . 8  |-  2  <  3
4341, 42gtneii 7516 . . . . . . 7  |-  3  =/=  2
44 neeq1 2264 . . . . . . 7  |-  ( x  =  3  ->  (
x  =/=  2  <->  3  =/=  2 ) )
4543, 44mpbiri 166 . . . . . 6  |-  ( x  =  3  ->  x  =/=  2 )
46 ifnefalse 3390 . . . . . 6  |-  ( x  =/=  2  ->  if ( x  =  2 ,  B ,  C )  =  C )
4745, 46syl 14 . . . . 5  |-  ( x  =  3  ->  if ( x  =  2 ,  B ,  C )  =  C )
4840, 47eqtrd 2117 . . . 4  |-  ( x  =  3  ->  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  =  C )
4935, 48eqeq12d 2099 . . 3  |-  ( x  =  3  ->  (
( F `  x
)  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C ) )  <->  ( F ` 
3 )  =  C ) )
5018, 19, 20, 23, 34, 49raltp 3481 . 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 182 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 103    /\ w3a 922    = wceq 1287    e. wcel 1436    =/= wne 2251   A.wral 2355   ifcif 3379   {ctp 3432   ` cfv 4977  (class class class)co 5606   1c1 7287    + caddc 7289   2c2 8399   3c3 8400   ZZcz 8675   ...cfz 9348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324  ax-cnex 7372  ax-resscn 7373  ax-1cn 7374  ax-1re 7375  ax-icn 7376  ax-addcl 7377  ax-addrcl 7378  ax-mulcl 7379  ax-addcom 7381  ax-addass 7383  ax-distr 7385  ax-i2m1 7386  ax-0lt1 7387  ax-0id 7389  ax-rnegex 7390  ax-cnre 7392  ax-pre-ltirr 7393  ax-pre-ltwlin 7394  ax-pre-lttrn 7395  ax-pre-apti 7396  ax-pre-ltadd 7397
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-if 3380  df-pw 3416  df-sn 3436  df-pr 3437  df-tp 3438  df-op 3439  df-uni 3636  df-int 3671  df-br 3820  df-opab 3874  df-mpt 3875  df-id 4092  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-res 4421  df-ima 4422  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-fv 4985  df-riota 5562  df-ov 5609  df-oprab 5610  df-mpt2 5611  df-pnf 7460  df-mnf 7461  df-xr 7462  df-ltxr 7463  df-le 7464  df-sub 7591  df-neg 7592  df-inn 8350  df-2 8408  df-3 8409  df-n0 8599  df-z 8676  df-uz 8944  df-fz 9349
This theorem is referenced by: (None)
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