Theorem fztpval 10490

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

Step	Hyp	Ref	Expression
1		1z 9670	. . . . 5 |- 1 e. ZZ
2		fztp 10485	. . . . 5 |- ( 1 e. ZZ -> ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
3	1, 2	ax-mp 5	. . . 4 |- ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
4		df-3 9364	. . . . . 6 |- 3 = ( 2 + 1 )
5		2cn 9375	. . . . . . 7 |- 2 e. CC
6		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
7	5, 6	addcomi 8470	. . . . . 6 |- ( 2 + 1 ) = ( 1 + 2 )
8	4, 7	eqtri 2259	. . . . 5 |- 3 = ( 1 + 2 )
9	8	oveq2i 6096	. . . 4 |- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) )
10		tpeq3 3799	. . . . . 6 |- ( 3 = ( 1 + 2 ) -> { 1 , 2 , 3 } = { 1 , 2 , ( 1 + 2 ) } )
11	8, 10	ax-mp 5	. . . . 5 |- { 1 , 2 , 3 } = { 1 , 2 , ( 1 + 2 ) }
12		df-2 9363	. . . . . 6 |- 2 = ( 1 + 1 )
13		tpeq2 3798	. . . . . 6 |- ( 2 = ( 1 + 1 ) -> { 1 , 2 , ( 1 + 2 ) } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
14	12, 13	ax-mp 5	. . . . 5 |- { 1 , 2 , ( 1 + 2 ) } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
15	11, 14	eqtri 2259	. . . 4 |- { 1 , 2 , 3 } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
16	3, 9, 15	3eqtr4i 2269	. . 3 |- ( 1 ... 3 ) = { 1 , 2 , 3 }
17	16	raleqi 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 8321	. . 3 |- 1 e. _V
19		2ex 9376	. . 3 |- 2 e. _V
20		3ex 9380	. . 3 |- 3 e. _V
21		fveq2 5695	. . . 4 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
22		iftrue 3645	. . . 4 |- ( x = 1 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = A )
23	21, 22	eqeq12d 2253	. . 3 |- ( x = 1 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 1 ) = A ) )
24		fveq2 5695	. . . 4 |- ( x = 2 -> ( F ` x ) = ( F ` 2 ) )
25		1re 8325	. . . . . . . 8 |- 1 e. RR
26		1lt2 9474	. . . . . . . 8 |- 1 < 2
27	25, 26	gtneii 8421	. . . . . . 7 |- 2 =/= 1
28		neeq1 2433	. . . . . . 7 |- ( x = 2 -> ( x =/= 1 <-> 2 =/= 1 ) )
29	27, 28	mpbiri 168	. . . . . 6 |- ( x = 2 -> x =/= 1 )
30		ifnefalse 3651	. . . . . 6 |- ( x =/= 1 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = if ( x = 2 , B , C ) )
31	29, 30	syl 14	. . . . 5 |- ( x = 2 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = if ( x = 2 , B , C ) )
32		iftrue 3645	. . . . 5 |- ( x = 2 -> if ( x = 2 , B , C ) = B )
33	31, 32	eqtrd 2271	. . . 4 |- ( x = 2 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = B )
34	24, 33	eqeq12d 2253	. . 3 |- ( x = 2 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 2 ) = B ) )
35		fveq2 5695	. . . 4 |- ( x = 3 -> ( F ` x ) = ( F ` 3 ) )
36		1lt3 9476	. . . . . . . 8 |- 1 < 3
37	25, 36	gtneii 8421	. . . . . . 7 |- 3 =/= 1
38		neeq1 2433	. . . . . . 7 |- ( x = 3 -> ( x =/= 1 <-> 3 =/= 1 ) )
39	37, 38	mpbiri 168	. . . . . 6 |- ( x = 3 -> x =/= 1 )
40	39, 30	syl 14	. . . . 5 |- ( x = 3 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = if ( x = 2 , B , C ) )
41		2re 9374	. . . . . . . 8 |- 2 e. RR
42		2lt3 9475	. . . . . . . 8 |- 2 < 3
43	41, 42	gtneii 8421	. . . . . . 7 |- 3 =/= 2
44		neeq1 2433	. . . . . . 7 |- ( x = 3 -> ( x =/= 2 <-> 3 =/= 2 ) )
45	43, 44	mpbiri 168	. . . . . 6 |- ( x = 3 -> x =/= 2 )
46		ifnefalse 3651	. . . . . 6 |- ( x =/= 2 -> if ( x = 2 , B , C ) = C )
47	45, 46	syl 14	. . . . 5 |- ( x = 3 -> if ( x = 2 , B , C ) = C )
48	40, 47	eqtrd 2271	. . . 4 |- ( x = 3 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = C )
49	35, 48	eqeq12d 2253	. . 3 |- ( x = 3 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 3 ) = C ) )
50	18, 19, 20, 23, 34, 49	raltp 3766	. 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 ) )
51	17, 50	bitri 184	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 ) )

This proof depends on syntax axioms:   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   ifcif 3638   {ctp 3711   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   2c2 9355   3c3 9356   ZZcz 9644   ...cfz 10411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412

This theorem is used by: (None)