Theorem fztpval 10158

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 9352	. . . . 5 |- 1 e. ZZ
2		fztp 10153	. . . . 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 9050	. . . . . 6 |- 3 = ( 2 + 1 )
5		2cn 9061	. . . . . . 7 |- 2 e. CC
6		ax-1cn 7972	. . . . . . 7 |- 1 e. CC
7	5, 6	addcomi 8170	. . . . . 6 |- ( 2 + 1 ) = ( 1 + 2 )
8	4, 7	eqtri 2217	. . . . 5 |- 3 = ( 1 + 2 )
9	8	oveq2i 5933	. . . 4 |- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) )
10		tpeq3 3710	. . . . . 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 9049	. . . . . 6 |- 2 = ( 1 + 1 )
13		tpeq2 3709	. . . . . 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 2217	. . . 4 |- { 1 , 2 , 3 } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
16	3, 9, 15	3eqtr4i 2227	. . 3 |- ( 1 ... 3 ) = { 1 , 2 , 3 }
17	16	raleqi 2697	. 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 8021	. . 3 |- 1 e. _V
19		2ex 9062	. . 3 |- 2 e. _V
20		3ex 9066	. . 3 |- 3 e. _V
21		fveq2 5558	. . . 4 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
22		iftrue 3566	. . . 4 |- ( x = 1 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = A )
23	21, 22	eqeq12d 2211	. . 3 |- ( x = 1 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 1 ) = A ) )
24		fveq2 5558	. . . 4 |- ( x = 2 -> ( F ` x ) = ( F ` 2 ) )
25		1re 8025	. . . . . . . 8 |- 1 e. RR
26		1lt2 9160	. . . . . . . 8 |- 1 < 2
27	25, 26	gtneii 8122	. . . . . . 7 |- 2 =/= 1
28		neeq1 2380	. . . . . . 7 |- ( x = 2 -> ( x =/= 1 <-> 2 =/= 1 ) )
29	27, 28	mpbiri 168	. . . . . 6 |- ( x = 2 -> x =/= 1 )
30		ifnefalse 3572	. . . . . 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 3566	. . . . 5 |- ( x = 2 -> if ( x = 2 , B , C ) = B )
33	31, 32	eqtrd 2229	. . . 4 |- ( x = 2 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = B )
34	24, 33	eqeq12d 2211	. . 3 |- ( x = 2 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 2 ) = B ) )
35		fveq2 5558	. . . 4 |- ( x = 3 -> ( F ` x ) = ( F ` 3 ) )
36		1lt3 9162	. . . . . . . 8 |- 1 < 3
37	25, 36	gtneii 8122	. . . . . . 7 |- 3 =/= 1
38		neeq1 2380	. . . . . . 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 9060	. . . . . . . 8 |- 2 e. RR
42		2lt3 9161	. . . . . . . 8 |- 2 < 3
43	41, 42	gtneii 8122	. . . . . . 7 |- 3 =/= 2
44		neeq1 2380	. . . . . . 7 |- ( x = 3 -> ( x =/= 2 <-> 3 =/= 2 ) )
45	43, 44	mpbiri 168	. . . . . 6 |- ( x = 3 -> x =/= 2 )
46		ifnefalse 3572	. . . . . 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 2229	. . . 4 |- ( x = 3 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = C )
49	35, 48	eqeq12d 2211	. . 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 3679	. 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 ) )

Syntax hints:   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   ifcif 3561   {ctp 3624   ` cfv 5258  (class class class)co 5922   1c1 7880   + caddc 7882   2c2 9041   3c3 9042   ZZcz 9326   ...cfz 10083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-3 9050  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084

This theorem is referenced by: (None)