![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fib6 | Structured version Visualization version GIF version |
Description: Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
Ref | Expression |
---|---|
fib6 | ⊢ (Fibci‘6) = 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5p1e6 12258 | . . 3 ⊢ (5 + 1) = 6 | |
2 | 1 | fveq2i 6842 | . 2 ⊢ (Fibci‘(5 + 1)) = (Fibci‘6) |
3 | 5nn 12197 | . . . 4 ⊢ 5 ∈ ℕ | |
4 | fibp1 32804 | . . . 4 ⊢ (5 ∈ ℕ → (Fibci‘(5 + 1)) = ((Fibci‘(5 − 1)) + (Fibci‘5))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (Fibci‘(5 + 1)) = ((Fibci‘(5 − 1)) + (Fibci‘5)) |
6 | 5cn 12199 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
7 | ax-1cn 11067 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 4cn 12196 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
9 | 4p1e5 12257 | . . . . . . . 8 ⊢ (4 + 1) = 5 | |
10 | 8, 7, 9 | addcomli 11305 | . . . . . . 7 ⊢ (1 + 4) = 5 |
11 | 6, 7, 8, 10 | subaddrii 11448 | . . . . . 6 ⊢ (5 − 1) = 4 |
12 | 11 | fveq2i 6842 | . . . . 5 ⊢ (Fibci‘(5 − 1)) = (Fibci‘4) |
13 | fib4 32807 | . . . . 5 ⊢ (Fibci‘4) = 3 | |
14 | 12, 13 | eqtri 2765 | . . . 4 ⊢ (Fibci‘(5 − 1)) = 3 |
15 | fib5 32808 | . . . 4 ⊢ (Fibci‘5) = 5 | |
16 | 14, 15 | oveq12i 7363 | . . 3 ⊢ ((Fibci‘(5 − 1)) + (Fibci‘5)) = (3 + 5) |
17 | 3cn 12192 | . . . 4 ⊢ 3 ∈ ℂ | |
18 | 5p3e8 12268 | . . . 4 ⊢ (5 + 3) = 8 | |
19 | 6, 17, 18 | addcomli 11305 | . . 3 ⊢ (3 + 5) = 8 |
20 | 5, 16, 19 | 3eqtri 2769 | . 2 ⊢ (Fibci‘(5 + 1)) = 8 |
21 | 2, 20 | eqtr3i 2767 | 1 ⊢ (Fibci‘6) = 8 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 1c1 11010 + caddc 11012 − cmin 11343 ℕcn 12111 3c3 12167 4c4 12168 5c5 12169 6c6 12170 8c8 12172 Fibcicfib 32799 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-n0 12372 df-xnn0 12444 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14184 df-word 14356 df-lsw 14404 df-concat 14412 df-s1 14437 df-substr 14486 df-pfx 14516 df-s2 14694 df-sseq 32787 df-fib 32800 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |