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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 33334 | . . 3 ⊢ Fibci = (〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1))))) | |
2 | 1 | fveq1i 6889 | . 2 ⊢ (Fibci‘0) = ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) |
3 | nn0ex 12474 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
5 | 0nn0 12483 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → 0 ∈ ℕ0) |
7 | 1nn0 12484 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ ℕ0) |
9 | 6, 8 | s2cld 14818 | . . . 4 ⊢ (⊤ → 〈“01”〉 ∈ Word ℕ0) |
10 | eqid 2733 | . . . 4 ⊢ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘〈“01”〉)))) = (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘〈“01”〉)))) | |
11 | fiblem 33335 | . . . . 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 12281 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | lbfzo0 13668 | . . . . . . 7 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
15 | 13, 14 | mpbir 230 | . . . . . 6 ⊢ 0 ∈ (0..^2) |
16 | s2len 14836 | . . . . . . 7 ⊢ (♯‘〈“01”〉) = 2 | |
17 | 16 | oveq2i 7415 | . . . . . 6 ⊢ (0..^(♯‘〈“01”〉)) = (0..^2) |
18 | 15, 17 | eleqtrri 2833 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘〈“01”〉)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → 0 ∈ (0..^(♯‘〈“01”〉))) |
20 | 4, 9, 10, 12, 19 | sseqfv1 33326 | . . 3 ⊢ (⊤ → ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) = (〈“01”〉‘0)) |
21 | 20 | mptru 1549 | . 2 ⊢ ((〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) = (〈“01”〉‘0) |
22 | s2fv0 14834 | . . 3 ⊢ (0 ∈ ℕ0 → (〈“01”〉‘0) = 0) | |
23 | 5, 22 | ax-mp 5 | . 2 ⊢ (〈“01”〉‘0) = 0 |
24 | 2, 21, 23 | 3eqtri 2765 | 1 ⊢ (Fibci‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3475 ∩ cin 3946 ↦ cmpt 5230 ◡ccnv 5674 “ cima 5678 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 0cc0 11106 1c1 11107 + caddc 11109 − cmin 11440 ℕcn 12208 2c2 12263 ℕ0cn0 12468 ℤ≥cuz 12818 ..^cfzo 13623 ♯chash 14286 Word cword 14460 〈“cs2 14788 seqstrcsseq 33320 Fibcicfib 33333 |
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 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-s2 14795 df-sseq 33321 df-fib 33334 |
This theorem is referenced by: fib2 33339 |
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