Theorem fztpval 10421

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 9605	. . . . 5 |- 1 e. ZZ
2		fztp 10416	. . . . 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 9299	. . . . . 6 |- 3 = ( 2 + 1 )
5		2cn 9310	. . . . . . 7 |- 2 e. CC
6		ax-1cn 8222	. . . . . . 7 |- 1 e. CC
7	5, 6	addcomi 8419	. . . . . 6 |- ( 2 + 1 ) = ( 1 + 2 )
8	4, 7	eqtri 2255	. . . . 5 |- 3 = ( 1 + 2 )
9	8	oveq2i 6063	. . . 4 |- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) )
10		tpeq3 3781	. . . . . 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 9298	. . . . . 6 |- 2 = ( 1 + 1 )
13		tpeq2 3780	. . . . . 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 2255	. . . 4 |- { 1 , 2 , 3 } = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
16	3, 9, 15	3eqtr4i 2265	. . 3 |- ( 1 ... 3 ) = { 1 , 2 , 3 }
17	16	raleqi 2747	. 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 8271	. . 3 |- 1 e. _V
19		2ex 9311	. . 3 |- 2 e. _V
20		3ex 9315	. . 3 |- 3 e. _V
21		fveq2 5672	. . . 4 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
22		iftrue 3629	. . . 4 |- ( x = 1 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = A )
23	21, 22	eqeq12d 2249	. . 3 |- ( x = 1 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 1 ) = A ) )
24		fveq2 5672	. . . 4 |- ( x = 2 -> ( F ` x ) = ( F ` 2 ) )
25		1re 8275	. . . . . . . 8 |- 1 e. RR
26		1lt2 9409	. . . . . . . 8 |- 1 < 2
27	25, 26	gtneii 8371	. . . . . . 7 |- 2 =/= 1
28		neeq1 2427	. . . . . . 7 |- ( x = 2 -> ( x =/= 1 <-> 2 =/= 1 ) )
29	27, 28	mpbiri 168	. . . . . 6 |- ( x = 2 -> x =/= 1 )
30		ifnefalse 3635	. . . . . 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 3629	. . . . 5 |- ( x = 2 -> if ( x = 2 , B , C ) = B )
33	31, 32	eqtrd 2267	. . . 4 |- ( x = 2 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = B )
34	24, 33	eqeq12d 2249	. . 3 |- ( x = 2 -> ( ( F ` x ) = if ( x = 1 , A , if ( x = 2 , B , C ) ) <-> ( F ` 2 ) = B ) )
35		fveq2 5672	. . . 4 |- ( x = 3 -> ( F ` x ) = ( F ` 3 ) )
36		1lt3 9411	. . . . . . . 8 |- 1 < 3
37	25, 36	gtneii 8371	. . . . . . 7 |- 3 =/= 1
38		neeq1 2427	. . . . . . 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 9309	. . . . . . . 8 |- 2 e. RR
42		2lt3 9410	. . . . . . . 8 |- 2 < 3
43	41, 42	gtneii 8371	. . . . . . 7 |- 3 =/= 2
44		neeq1 2427	. . . . . . 7 |- ( x = 3 -> ( x =/= 2 <-> 3 =/= 2 ) )
45	43, 44	mpbiri 168	. . . . . 6 |- ( x = 3 -> x =/= 2 )
46		ifnefalse 3635	. . . . . 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 2267	. . . 4 |- ( x = 3 -> if ( x = 1 , A , if ( x = 2 , B , C ) ) = C )
49	35, 48	eqeq12d 2249	. . 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 3748	. 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 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   ifcif 3622   {ctp 3693   ` cfv 5354  (class class class)co 6052   1c1 8130   + caddc 8132   2c2 9290   3c3 9291   ZZcz 9579   ...cfz 10345

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-2 9298  df-3 9299  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346

This theorem is referenced by: (None)