Theorem fztpval 7688

Description: Two ways of defining the first three values of a sequence on NN.

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 7368	. . . . 5 |- 1 e. ZZ
2		fztp 7686	. . . . 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 7155	. . . . . 6 |- 3 = (2 + 1)
5		2cn 7164	. . . . . . 7 |- 2 e. CC
6		ax1cn 6422	. . . . . . 7 |- 1 e. CC
7	5, 6	addcomi 6475	. . . . . 6 |- (2 + 1) = (1 + 2)
8	4, 7	eqtri 1908	. . . . 5 |- 3 = (1 + 2)
9	8	opreq2i 4893	. . . 4 |- (1...3) = (1...(1 + 2))
10		tpeq3 3102	. . . . . 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 7154	. . . . . 6 |- 2 = (1 + 1)
13		tpeq2 3101	. . . . . 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 1908	. . . 4 |- {1, 2, 3} = {1, (1 + 1), (1 + 2)}
16	3, 9, 15	3eqtr4i 1921	. . 3 |- (1...3) = {1, 2, 3}
17		raleq 2266	. . 3 |- ((1...3) = {1, 2, 3} -> (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	16, 17	ax-mp 7	. 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)))
19	6	elisseti 2301	. . 3 |- 1 e. _V
20	5	elisseti 2301	. . 3 |- 2 e. _V
21		3re 7165	. . . 4 |- 3 e. RR
22	21	elisseti 2301	. . 3 |- 3 e. _V
23	19, 20, 22	raltp 3083	. 2 |- (A.x e. {1, 2, 3} (F` x) = if(x = 1, A, if(x = 2, B, C)) <-> ([1 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) /\ [2 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) /\ [3 / x](F` x) = if(x = 1, A, if(x = 2, B, C))))
24		fveq2 4681	. . . . 5 |- (x = 1 -> (F` x) = (F` 1))
25		iftrue 2989	. . . . 5 |- (x = 1 -> if(x = 1, A, if(x = 2, B, C)) = A)
26	24, 25	eqeq12d 1899	. . . 4 |- (x = 1 -> ((F` x) = if(x = 1, A, if(x = 2, B, C)) <-> (F` 1) = A))
27	19, 26	sbcie 2485	. . 3 |- ([1 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) <-> (F` 1) = A)
28		fveq2 4681	. . . . 5 |- (x = 2 -> (F` x) = (F` 2))
29		1lt2 7212	. . . . . . . . . 10 |- 1 < 2
30		1re 6598	. . . . . . . . . . 11 |- 1 e. RR
31		2re 7163	. . . . . . . . . . 11 |- 2 e. RR
32	30, 31	ltnei 6758	. . . . . . . . . 10 |- (1 < 2 -> 2 =/= 1)
33	29, 32	ax-mp 7	. . . . . . . . 9 |- 2 =/= 1
34		pm13.181 2086	. . . . . . . . 9 |- ((x = 2 /\ 2 =/= 1) -> x =/= 1)
35	33, 34	mpan2 760	. . . . . . . 8 |- (x = 2 -> x =/= 1)
36		df-ne 2019	. . . . . . . 8 |- (x =/= 1 <-> -. x = 1)
37	35, 36	sylib 215	. . . . . . 7 |- (x = 2 -> -. x = 1)
38		iffalse 2991	. . . . . . 7 |- (-. x = 1 -> if(x = 1, A, if(x = 2, B, C)) = if(x = 2, B, C))
39	37, 38	syl 12	. . . . . 6 |- (x = 2 -> if(x = 1, A, if(x = 2, B, C)) = if(x = 2, B, C))
40		iftrue 2989	. . . . . 6 |- (x = 2 -> if(x = 2, B, C) = B)
41	39, 40	eqtrd 1925	. . . . 5 |- (x = 2 -> if(x = 1, A, if(x = 2, B, C)) = B)
42	28, 41	eqeq12d 1899	. . . 4 |- (x = 2 -> ((F` x) = if(x = 1, A, if(x = 2, B, C)) <-> (F` 2) = B))
43	20, 42	sbcie 2485	. . 3 |- ([2 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) <-> (F` 2) = B)
44		fveq2 4681	. . . . 5 |- (x = 3 -> (F` x) = (F` 3))
45		1lt3 7214	. . . . . . . . . 10 |- 1 < 3
46	30, 21	ltnei 6758	. . . . . . . . . 10 |- (1 < 3 -> 3 =/= 1)
47	45, 46	ax-mp 7	. . . . . . . . 9 |- 3 =/= 1
48		pm13.181 2086	. . . . . . . . 9 |- ((x = 3 /\ 3 =/= 1) -> x =/= 1)
49	47, 48	mpan2 760	. . . . . . . 8 |- (x = 3 -> x =/= 1)
50	49, 36	sylib 215	. . . . . . 7 |- (x = 3 -> -. x = 1)
51	50, 38	syl 12	. . . . . 6 |- (x = 3 -> if(x = 1, A, if(x = 2, B, C)) = if(x = 2, B, C))
52		2lt3 7213	. . . . . . . . . 10 |- 2 < 3
53	31, 21	ltnei 6758	. . . . . . . . . 10 |- (2 < 3 -> 3 =/= 2)
54	52, 53	ax-mp 7	. . . . . . . . 9 |- 3 =/= 2
55		pm13.181 2086	. . . . . . . . 9 |- ((x = 3 /\ 3 =/= 2) -> x =/= 2)
56	54, 55	mpan2 760	. . . . . . . 8 |- (x = 3 -> x =/= 2)
57		df-ne 2019	. . . . . . . 8 |- (x =/= 2 <-> -. x = 2)
58	56, 57	sylib 215	. . . . . . 7 |- (x = 3 -> -. x = 2)
59		iffalse 2991	. . . . . . 7 |- (-. x = 2 -> if(x = 2, B, C) = C)
60	58, 59	syl 12	. . . . . 6 |- (x = 3 -> if(x = 2, B, C) = C)
61	51, 60	eqtrd 1925	. . . . 5 |- (x = 3 -> if(x = 1, A, if(x = 2, B, C)) = C)
62	44, 61	eqeq12d 1899	. . . 4 |- (x = 3 -> ((F` x) = if(x = 1, A, if(x = 2, B, C)) <-> (F` 3) = C))
63	22, 62	sbcie 2485	. . 3 |- ([3 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) <-> (F` 3) = C)
64	27, 43, 63	3anbi123i 1056	. 2 |- (([1 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) /\ [2 / x](F` x) = if(x = 1, A, if(x = 2, B, C)) /\ [3 / x](F` x) = if(x = 1, A, if(x = 2, B, C))) <-> ((F` 1) = A /\ (F` 2) = B /\ (F` 3) = C))
65	18, 23, 64	3bitri 194	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:  -. wn 2   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534   =/= wne 2017  A.wral 2105  ifcif 2982  {ctp 3051   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389  ZZcz 6451   < clt 6653  2c2 7145  3c3 7146  ...cfz 7637

This theorem is referenced by:  stb3el 16737  stb3cl 16738  stb3val1 16739  stb3val2 16740  stb3val3 16741  stb3xpl 16743

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-3 7155  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638

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