Theorem fztpval 9100

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 8377	. . . . 5 |- 1 e. ZZ
2		fztp 9095	. . . . 5 |- ( 1 e. ZZ -> ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
3	1, 2	ax-mp 7	. . . 4 |- ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
4		df-3 8099	. . . . . 6 |- 3 = ( 2 + 1 )
5		2cn 8110	. . . . . . 7 |- 2 e. CC
6		ax-1cn 7069	. . . . . . 7 |- 1 e. CC
7	5, 6	addcomi 7252	. . . . . 6 |- ( 2 + 1 ) = ( 1 + 2 )
8	4, 7	eqtri 2101	. . . . 5 |- 3 = ( 1 + 2 )
9	8	oveq2i 5543	. . . 4 |- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) )
10		tpeq3 3480	. . . . . 6 |- ( 3 = ( 1 + 2 ) -> { 1 , 2 , 3 } = { 1 , 2 , ( 1 + 2 ) } )
11	8, 10	ax-mp 7	. . . . 5 |- { 1 , 2 , 3 } = { 1 , 2 , ( 1 + 2 ) }
12		df-2 8098	. . . . . 6 |- 2 = ( 1 + 1 )
13		tpeq2 3479	. . . . . 6 |- ( 2 = ( 1 + 1 ) -> { 1 , 2 , ( 1 + 2 ) } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
14	12, 13	ax-mp 7	. . . . 5 |- { 1 , 2 , ( 1 + 2 ) } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
15	11, 14	eqtri 2101	. . . 4 |- { 1 , 2 , 3 } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
16	3, 9, 15	3eqtr4i 2111	. . 3 |- ( 1 ... 3 ) = { 1 , 2 , 3 }
17	16	raleqi 2553	. 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 7114	. . 3 |- 1 e. _V
19		2ex 8111	. . 3 |- 2 e. _V
20		3ex 8115	. . 3 |- 3 e. _V
21		fveq2 5198	. . . 4 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
22		iftrue 3356	. . . 4 |- ( x = 1 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = A )
23	21, 22	eqeq12d 2095	. . 3 |- ( x = 1 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 1 ) = A ) )
24		fveq2 5198	. . . 4 |- ( x = 2 -> ( F ` x ) = ( F ` 2 ) )
25		1re 7118	. . . . . . . 8 |- 1 e. RR
26		1lt2 8201	. . . . . . . 8 |- 1 < 2
27	25, 26	gtneii 7206	. . . . . . 7 |- 2 =/= 1
28		neeq1 2258	. . . . . . 7 |- ( x = 2 -> ( x =/= 1 <-> 2 =/= 1 ) )
29	27, 28	mpbiri 166	. . . . . 6 |- ( x = 2 -> x =/= 1 )
30		ifnefalse 3362	. . . . . 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 3356	. . . . 5 |- ( x = 2 -> if ( x = 2 , B , C ) = B )
33	31, 32	eqtrd 2113	. . . 4 |- ( x = 2 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = B )
34	24, 33	eqeq12d 2095	. . 3 |- ( x = 2 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 2 ) = B ) )
35		fveq2 5198	. . . 4 |- ( x = 3 -> ( F ` x ) = ( F ` 3 ) )
36		1lt3 8203	. . . . . . . 8 |- 1 < 3
37	25, 36	gtneii 7206	. . . . . . 7 |- 3 =/= 1
38		neeq1 2258	. . . . . . 7 |- ( x = 3 -> ( x =/= 1 <-> 3 =/= 1 ) )
39	37, 38	mpbiri 166	. . . . . 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 8109	. . . . . . . 8 |- 2 e. RR
42		2lt3 8202	. . . . . . . 8 |- 2 < 3
43	41, 42	gtneii 7206	. . . . . . 7 |- 3 =/= 2
44		neeq1 2258	. . . . . . 7 |- ( x = 3 -> ( x =/= 2 <-> 3 =/= 2 ) )
45	43, 44	mpbiri 166	. . . . . 6 |- ( x = 3 -> x =/= 2 )
46		ifnefalse 3362	. . . . . 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 2113	. . . 4 |- ( x = 3 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = C )
49	35, 48	eqeq12d 2095	. . 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 3449	. 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 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 ) )

Syntax hints:   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   =/= wne 2245   A.wral 2348   ifcif 3351   {ctp 3400   ` cfv 4922  (class class class)co 5532   1c1 6982   + caddc 6984   2c2 8089   3c3 8090   ZZcz 8351   ...cfz 9029

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-tp 3406  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-2 8098  df-3 8099  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by: (None)