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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fib1 | Structured version Visualization version GIF version |
Description: Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
Ref | Expression |
---|---|
fib1 | ⊢ (Fibci‘1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fib 33037 | . . 3 ⊢ Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1))))) | |
2 | 1 | fveq1i 6848 | . 2 ⊢ (Fibci‘1) = ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘1) |
3 | nn0ex 12426 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
5 | 0nn0 12435 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → 0 ∈ ℕ0) |
7 | 1nn0 12436 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ ℕ0) |
9 | 6, 8 | s2cld 14767 | . . . 4 ⊢ (⊤ → ⟨“01”⟩ ∈ Word ℕ0) |
10 | eqid 2737 | . . . 4 ⊢ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩)))) = (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩)))) | |
11 | fiblem 33038 | . . . . 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 12233 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 1lt2 12331 | . . . . . . 7 ⊢ 1 < 2 | |
15 | elfzo0 13620 | . . . . . . 7 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2)) | |
16 | 7, 13, 14, 15 | mpbir3an 1342 | . . . . . 6 ⊢ 1 ∈ (0..^2) |
17 | s2len 14785 | . . . . . . 7 ⊢ (♯‘⟨“01”⟩) = 2 | |
18 | 17 | oveq2i 7373 | . . . . . 6 ⊢ (0..^(♯‘⟨“01”⟩)) = (0..^2) |
19 | 16, 18 | eleqtrri 2837 | . . . . 5 ⊢ 1 ∈ (0..^(♯‘⟨“01”⟩)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ (0..^(♯‘⟨“01”⟩))) |
21 | 4, 9, 10, 12, 20 | sseqfv1 33029 | . . 3 ⊢ (⊤ → ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘1) = (⟨“01”⟩‘1)) |
22 | 21 | mptru 1549 | . 2 ⊢ ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘1) = (⟨“01”⟩‘1) |
23 | s2fv1 14784 | . . 3 ⊢ (1 ∈ ℕ0 → (⟨“01”⟩‘1) = 1) | |
24 | 7, 23 | ax-mp 5 | . 2 ⊢ (⟨“01”⟩‘1) = 1 |
25 | 2, 22, 24 | 3eqtri 2769 | 1 ⊢ (Fibci‘1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3448 ∩ cin 3914 class class class wbr 5110 ↦ cmpt 5193 ◡ccnv 5637 “ cima 5641 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 − cmin 11392 ℕcn 12160 2c2 12215 ℕ0cn0 12420 ℤ≥cuz 12770 ..^cfzo 13574 ♯chash 14237 Word cword 14409 ⟨“cs2 14737 seqstrcsseq 33023 Fibcicfib 33036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-word 14410 df-lsw 14458 df-concat 14466 df-s1 14491 df-s2 14744 df-sseq 33024 df-fib 33037 |
This theorem is referenced by: fib2 33042 fib3 33043 |
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