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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 34011 | . . 3 ⊢ Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1))))) | |
2 | 1 | fveq1i 6892 | . 2 ⊢ (Fibci‘1) = ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘1) |
3 | nn0ex 12502 | . . . . 5 ⊢ ℕ0 ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
5 | 0nn0 12511 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → 0 ∈ ℕ0) |
7 | 1nn0 12512 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ ℕ0) |
9 | 6, 8 | s2cld 14848 | . . . 4 ⊢ (⊤ → ⟨“01”⟩ ∈ Word ℕ0) |
10 | eqid 2728 | . . . 4 ⊢ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩)))) = (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩)))) | |
11 | fiblem 34012 | . . . . 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 12309 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 1lt2 12407 | . . . . . . 7 ⊢ 1 < 2 | |
15 | elfzo0 13699 | . . . . . . 7 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2)) | |
16 | 7, 13, 14, 15 | mpbir3an 1339 | . . . . . 6 ⊢ 1 ∈ (0..^2) |
17 | s2len 14866 | . . . . . . 7 ⊢ (♯‘⟨“01”⟩) = 2 | |
18 | 17 | oveq2i 7425 | . . . . . 6 ⊢ (0..^(♯‘⟨“01”⟩)) = (0..^2) |
19 | 16, 18 | eleqtrri 2828 | . . . . 5 ⊢ 1 ∈ (0..^(♯‘⟨“01”⟩)) |
20 | 19 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ (0..^(♯‘⟨“01”⟩))) |
21 | 4, 9, 10, 12, 20 | sseqfv1 34003 | . . 3 ⊢ (⊤ → ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘1) = (⟨“01”⟩‘1)) |
22 | 21 | mptru 1541 | . 2 ⊢ ((⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))‘1) = (⟨“01”⟩‘1) |
23 | s2fv1 14865 | . . 3 ⊢ (1 ∈ ℕ0 → (⟨“01”⟩‘1) = 1) | |
24 | 7, 23 | ax-mp 5 | . 2 ⊢ (⟨“01”⟩‘1) = 1 |
25 | 2, 22, 24 | 3eqtri 2760 | 1 ⊢ (Fibci‘1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3470 ∩ cin 3944 class class class wbr 5142 ↦ cmpt 5225 ◡ccnv 5671 “ cima 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 + caddc 11135 < clt 11272 − cmin 11468 ℕcn 12236 2c2 12291 ℕ0cn0 12496 ℤ≥cuz 12846 ..^cfzo 13653 ♯chash 14315 Word cword 14490 ⟨“cs2 14818 seqstrcsseq 33997 Fibcicfib 34010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-n0 12497 df-xnn0 12569 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-word 14491 df-lsw 14539 df-concat 14547 df-s1 14572 df-s2 14825 df-sseq 33998 df-fib 34011 |
This theorem is referenced by: fib2 34016 fib3 34017 |
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