Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fib0 | Structured version Visualization version GIF version |
Description: Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
Ref | Expression |
---|---|
fib0 | ⊢ (Fibci‘0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fib 29786 | . . 3 ⊢ Fibci = (〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1))))) | |
2 | 1 | fveq1i 6104 | . 2 ⊢ (Fibci‘0) = ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))))‘0) |
3 | nn0ex 11175 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
5 | 0nn0 11184 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → 0 ∈ ℕ0) |
7 | 1nn0 11185 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ ℕ0) |
9 | 6, 8 | s2cld 13466 | . . . 4 ⊢ (⊤ → 〈“01”〉 ∈ Word ℕ0) |
10 | eqid 2610 | . . . 4 ⊢ (Word ℕ0 ∩ (◡# “ (ℤ≥‘(#‘〈“01”〉)))) = (Word ℕ0 ∩ (◡# “ (ℤ≥‘(#‘〈“01”〉)))) | |
11 | fiblem 29787 | . . . . 5 ⊢ (𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))):(Word ℕ0 ∩ (◡# “ (ℤ≥‘(#‘〈“01”〉))))⟶ℕ0 | |
12 | 11 | a1i 11 | . . . 4 ⊢ (⊤ → (𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))):(Word ℕ0 ∩ (◡# “ (ℤ≥‘(#‘〈“01”〉))))⟶ℕ0) |
13 | 2nn 11062 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | lbfzo0 12375 | . . . . . . 7 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
15 | 13, 14 | mpbir 220 | . . . . . 6 ⊢ 0 ∈ (0..^2) |
16 | s2len 13484 | . . . . . . 7 ⊢ (#‘〈“01”〉) = 2 | |
17 | 16 | oveq2i 6560 | . . . . . 6 ⊢ (0..^(#‘〈“01”〉)) = (0..^2) |
18 | 15, 17 | eleqtrri 2687 | . . . . 5 ⊢ 0 ∈ (0..^(#‘〈“01”〉)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → 0 ∈ (0..^(#‘〈“01”〉))) |
20 | 4, 9, 10, 12, 19 | sseqfv1 29778 | . . 3 ⊢ (⊤ → ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))))‘0) = (〈“01”〉‘0)) |
21 | 20 | trud 1484 | . 2 ⊢ ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡# “ (ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1)))))‘0) = (〈“01”〉‘0) |
22 | s2fv0 13482 | . . 3 ⊢ (0 ∈ ℕ0 → (〈“01”〉‘0) = 0) | |
23 | 5, 22 | ax-mp 5 | . 2 ⊢ (〈“01”〉‘0) = 0 |
24 | 2, 21, 23 | 3eqtri 2636 | 1 ⊢ (Fibci‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℤ≥cuz 11563 ..^cfzo 12334 #chash 12979 Word cword 13146 〈“cs2 13437 seqstrcsseq 29772 Fibcicfib 29785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-s2 13444 df-sseq 29773 df-fib 29786 |
This theorem is referenced by: fib2 29791 |
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