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 31655 | . . 3 ⊢ Fibci = (〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1))))) | |
2 | 1 | fveq1i 6670 | . 2 ⊢ (Fibci‘0) = ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) |
3 | nn0ex 11902 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
5 | 0nn0 11911 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → 0 ∈ ℕ0) |
7 | 1nn0 11912 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ ℕ0) |
9 | 6, 8 | s2cld 14232 | . . . 4 ⊢ (⊤ → 〈“01”〉 ∈ Word ℕ0) |
10 | eqid 2821 | . . . 4 ⊢ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘〈“01”〉)))) = (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘〈“01”〉)))) | |
11 | fiblem 31656 | . . . . 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 11709 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | lbfzo0 13076 | . . . . . . 7 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
15 | 13, 14 | mpbir 233 | . . . . . 6 ⊢ 0 ∈ (0..^2) |
16 | s2len 14250 | . . . . . . 7 ⊢ (♯‘〈“01”〉) = 2 | |
17 | 16 | oveq2i 7166 | . . . . . 6 ⊢ (0..^(♯‘〈“01”〉)) = (0..^2) |
18 | 15, 17 | eleqtrri 2912 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“01”〉)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → 0 ∈ (0..^(♯‘〈“01”〉))) |
20 | 4, 9, 10, 12, 19 | sseqfv1 31647 | . . 3 ⊢ (⊤ → ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) = (〈“01”〉‘0)) |
21 | 20 | mptru 1540 | . 2 ⊢ ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) = (〈“01”〉‘0) |
22 | s2fv0 14248 | . . 3 ⊢ (0 ∈ ℕ0 → (〈“01”〉‘0) = 0) | |
23 | 5, 22 | ax-mp 5 | . 2 ⊢ (〈“01”〉‘0) = 0 |
24 | 2, 21, 23 | 3eqtri 2848 | 1 ⊢ (Fibci‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ↦ cmpt 5145 ◡ccnv 5553 “ cima 5557 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 + caddc 10539 − cmin 10869 ℕcn 11637 2c2 11691 ℕ0cn0 11896 ℤ≥cuz 12242 ..^cfzo 13032 ♯chash 13689 Word cword 13860 〈“cs2 14202 seqstrcsseq 31641 Fibcicfib 31654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-s2 14209 df-sseq 31642 df-fib 31655 |
This theorem is referenced by: fib2 31660 |
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