Theorem fztpval 10152

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 9346	. . . . 5 |- 1 e. ZZ
2		fztp 10147	. . . . 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 9044	. . . . . 6 |- 3 = ( 2 + 1 )
5		2cn 9055	. . . . . . 7 |- 2 e. CC
6		ax-1cn 7967	. . . . . . 7 |- 1 e. CC
7	5, 6	addcomi 8165	. . . . . 6 |- ( 2 + 1 ) = ( 1 + 2 )
8	4, 7	eqtri 2214	. . . . 5 |- 3 = ( 1 + 2 )
9	8	oveq2i 5930	. . . 4 |- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) )
10		tpeq3 3707	. . . . . 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 9043	. . . . . 6 |- 2 = ( 1 + 1 )
13		tpeq2 3706	. . . . . 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 2214	. . . 4 |- { 1 , 2 , 3 } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
16	3, 9, 15	3eqtr4i 2224	. . 3 |- ( 1 ... 3 ) = { 1 , 2 , 3 }
17	16	raleqi 2694	. 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 8016	. . 3 |- 1 e. _V
19		2ex 9056	. . 3 |- 2 e. _V
20		3ex 9060	. . 3 |- 3 e. _V
21		fveq2 5555	. . . 4 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
22		iftrue 3563	. . . 4 |- ( x = 1 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = A )
23	21, 22	eqeq12d 2208	. . 3 |- ( x = 1 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 1 ) = A ) )
24		fveq2 5555	. . . 4 |- ( x = 2 -> ( F ` x ) = ( F ` 2 ) )
25		1re 8020	. . . . . . . 8 |- 1 e. RR
26		1lt2 9154	. . . . . . . 8 |- 1 < 2
27	25, 26	gtneii 8117	. . . . . . 7 |- 2 =/= 1
28		neeq1 2377	. . . . . . 7 |- ( x = 2 -> ( x =/= 1 <-> 2 =/= 1 ) )
29	27, 28	mpbiri 168	. . . . . 6 |- ( x = 2 -> x =/= 1 )
30		ifnefalse 3569	. . . . . 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 3563	. . . . 5 |- ( x = 2 -> if ( x = 2 , B , C ) = B )
33	31, 32	eqtrd 2226	. . . 4 |- ( x = 2 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = B )
34	24, 33	eqeq12d 2208	. . 3 |- ( x = 2 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 2 ) = B ) )
35		fveq2 5555	. . . 4 |- ( x = 3 -> ( F ` x ) = ( F ` 3 ) )
36		1lt3 9156	. . . . . . . 8 |- 1 < 3
37	25, 36	gtneii 8117	. . . . . . 7 |- 3 =/= 1
38		neeq1 2377	. . . . . . 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 9054	. . . . . . . 8 |- 2 e. RR
42		2lt3 9155	. . . . . . . 8 |- 2 < 3
43	41, 42	gtneii 8117	. . . . . . 7 |- 3 =/= 2
44		neeq1 2377	. . . . . . 7 |- ( x = 3 -> ( x =/= 2 <-> 3 =/= 2 ) )
45	43, 44	mpbiri 168	. . . . . 6 |- ( x = 3 -> x =/= 2 )
46		ifnefalse 3569	. . . . . 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 2226	. . . 4 |- ( x = 3 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = C )
49	35, 48	eqeq12d 2208	. . 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 3676	. 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 2164   =/= wne 2364   A.wral 2472   ifcif 3558   {ctp 3621   ` cfv 5255  (class class class)co 5919   1c1 7875   + caddc 7877   2c2 9035   3c3 9036   ZZcz 9320   ...cfz 10077

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-3 9044  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078

This theorem is referenced by: (None)