Theorem fztpval 10069

Description: Two ways of defining the first three values of a sequence on ℕ. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
fztpval	⊢ (∀𝑥 ∈ (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 9268	. . . . 5 ⊢ 1 ∈ ℤ
2		fztp 10064	. . . . 5 ⊢ (1 ∈ ℤ → (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 8968	. . . . . 6 ⊢ 3 = (2 + 1)
5		2cn 8979	. . . . . . 7 ⊢ 2 ∈ ℂ
6		ax-1cn 7895	. . . . . . 7 ⊢ 1 ∈ ℂ
7	5, 6	addcomi 8091	. . . . . 6 ⊢ (2 + 1) = (1 + 2)
8	4, 7	eqtri 2198	. . . . 5 ⊢ 3 = (1 + 2)
9	8	oveq2i 5880	. . . 4 ⊢ (1...3) = (1...(1 + 2))
10		tpeq3 3679	. . . . . 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 8967	. . . . . 6 ⊢ 2 = (1 + 1)
13		tpeq2 3678	. . . . . 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 2198	. . . 4 ⊢ {1, 2, 3} = {1, (1 + 1), (1 + 2)}
16	3, 9, 15	3eqtr4i 2208	. . 3 ⊢ (1...3) = {1, 2, 3}
17	16	raleqi 2676	. 2 ⊢ (∀𝑥 ∈ (1...3)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ ∀𝑥 ∈ {1, 2, 3} (𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)))
18		1ex 7943	. . 3 ⊢ 1 ∈ V
19		2ex 8980	. . 3 ⊢ 2 ∈ V
20		3ex 8984	. . 3 ⊢ 3 ∈ V
21		fveq2 5511	. . . 4 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1))
22		iftrue 3539	. . . 4 ⊢ (𝑥 = 1 → if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) = 𝐴)
23	21, 22	eqeq12d 2192	. . 3 ⊢ (𝑥 = 1 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ (𝐹‘1) = 𝐴))
24		fveq2 5511	. . . 4 ⊢ (𝑥 = 2 → (𝐹‘𝑥) = (𝐹‘2))
25		1re 7947	. . . . . . . 8 ⊢ 1 ∈ ℝ
26		1lt2 9077	. . . . . . . 8 ⊢ 1 < 2
27	25, 26	gtneii 8043	. . . . . . 7 ⊢ 2 ≠ 1
28		neeq1 2360	. . . . . . 7 ⊢ (𝑥 = 2 → (𝑥 ≠ 1 ↔ 2 ≠ 1))
29	27, 28	mpbiri 168	. . . . . 6 ⊢ (𝑥 = 2 → 𝑥 ≠ 1)
30		ifnefalse 3545	. . . . . 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 3539	. . . . 5 ⊢ (𝑥 = 2 → if(𝑥 = 2, 𝐵, 𝐶) = 𝐵)
33	31, 32	eqtrd 2210	. . . 4 ⊢ (𝑥 = 2 → if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) = 𝐵)
34	24, 33	eqeq12d 2192	. . 3 ⊢ (𝑥 = 2 → ((𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ (𝐹‘2) = 𝐵))
35		fveq2 5511	. . . 4 ⊢ (𝑥 = 3 → (𝐹‘𝑥) = (𝐹‘3))
36		1lt3 9079	. . . . . . . 8 ⊢ 1 < 3
37	25, 36	gtneii 8043	. . . . . . 7 ⊢ 3 ≠ 1
38		neeq1 2360	. . . . . . 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 8978	. . . . . . . 8 ⊢ 2 ∈ ℝ
42		2lt3 9078	. . . . . . . 8 ⊢ 2 < 3
43	41, 42	gtneii 8043	. . . . . . 7 ⊢ 3 ≠ 2
44		neeq1 2360	. . . . . . 7 ⊢ (𝑥 = 3 → (𝑥 ≠ 2 ↔ 3 ≠ 2))
45	43, 44	mpbiri 168	. . . . . 6 ⊢ (𝑥 = 3 → 𝑥 ≠ 2)
46		ifnefalse 3545	. . . . . 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 2210	. . . 4 ⊢ (𝑥 = 3 → if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) = 𝐶)
49	35, 48	eqeq12d 2192	. . 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 3648	. 2 ⊢ (∀𝑥 ∈ {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 ⊢ (∀𝑥 ∈ (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 978   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ifcif 3534  {ctp 3593  ‘cfv 5212  (class class class)co 5869  1c1 7803   + caddc 7805  2c2 8959  3c3 8960  ℤcz 9242  ...cfz 9995

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-tp 3599  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-2 8967  df-3 8968  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996

This theorem is referenced by: (None)