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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 33391 | . . 3 ⊢ Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1))))) | |
2 | 1 | fveq1i 6892 | . 2 ⊢ (Fibci‘0) = ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) |
3 | nn0ex 12477 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
5 | 0nn0 12486 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → 0 ∈ ℕ0) |
7 | 1nn0 12487 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ ℕ0) |
9 | 6, 8 | s2cld 14821 | . . . 4 ⊢ (⊤ → ⟨“01”⟩ ∈ Word ℕ0) |
10 | eqid 2732 | . . . 4 ⊢ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩)))) = (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩)))) | |
11 | fiblem 33392 | . . . . 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 12284 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | lbfzo0 13671 | . . . . . . 7 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ) | |
15 | 13, 14 | mpbir 230 | . . . . . 6 ⊢ 0 ∈ (0..^2) |
16 | s2len 14839 | . . . . . . 7 ⊢ (♯‘⟨“01”⟩) = 2 | |
17 | 16 | oveq2i 7419 | . . . . . 6 ⊢ (0..^(♯‘⟨“01”⟩)) = (0..^2) |
18 | 15, 17 | eleqtrri 2832 | . . . . 5 ⊢ 0 ∈ (0..^(♯‘⟨“01”⟩)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → 0 ∈ (0..^(♯‘⟨“01”⟩))) |
20 | 4, 9, 10, 12, 19 | sseqfv1 33383 | . . 3 ⊢ (⊤ → ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) = (⟨“01”⟩‘0)) |
21 | 20 | mptru 1548 | . 2 ⊢ ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘0) = (⟨“01”⟩‘0) |
22 | s2fv0 14837 | . . 3 ⊢ (0 ∈ ℕ0 → (⟨“01”⟩‘0) = 0) | |
23 | 5, 22 | ax-mp 5 | . 2 ⊢ (⟨“01”⟩‘0) = 0 |
24 | 2, 21, 23 | 3eqtri 2764 | 1 ⊢ (Fibci‘0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 Vcvv 3474 ∩ cin 3947 ↦ cmpt 5231 ◡ccnv 5675 “ cima 5679 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 0cc0 11109 1c1 11110 + caddc 11112 − cmin 11443 ℕcn 12211 2c2 12266 ℕ0cn0 12471 ℤ≥cuz 12821 ..^cfzo 13626 ♯chash 14289 Word cword 14463 ⟨“cs2 14791 seqstrcsseq 33377 Fibcicfib 33390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-word 14464 df-lsw 14512 df-concat 14520 df-s1 14545 df-s2 14798 df-sseq 33378 df-fib 33391 |
This theorem is referenced by: fib2 33396 |
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