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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fib5 | Structured version Visualization version GIF version |
Description: Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
Ref | Expression |
---|---|
fib5 | ⊢ (Fibci‘5) = 5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4p1e5 11466 | . . 3 ⊢ (4 + 1) = 5 | |
2 | 1 | fveq2i 6414 | . 2 ⊢ (Fibci‘(4 + 1)) = (Fibci‘5) |
3 | 4nn 11397 | . . . 4 ⊢ 4 ∈ ℕ | |
4 | fibp1 30980 | . . . 4 ⊢ (4 ∈ ℕ → (Fibci‘(4 + 1)) = ((Fibci‘(4 − 1)) + (Fibci‘4))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (Fibci‘(4 + 1)) = ((Fibci‘(4 − 1)) + (Fibci‘4)) |
6 | 4cn 11399 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
7 | ax-1cn 10282 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 3cn 11394 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
9 | 3p1e4 11465 | . . . . . . . 8 ⊢ (3 + 1) = 4 | |
10 | 8, 7, 9 | addcomli 10518 | . . . . . . 7 ⊢ (1 + 3) = 4 |
11 | 6, 7, 8, 10 | subaddrii 10662 | . . . . . 6 ⊢ (4 − 1) = 3 |
12 | 11 | fveq2i 6414 | . . . . 5 ⊢ (Fibci‘(4 − 1)) = (Fibci‘3) |
13 | fib3 30982 | . . . . 5 ⊢ (Fibci‘3) = 2 | |
14 | 12, 13 | eqtri 2821 | . . . 4 ⊢ (Fibci‘(4 − 1)) = 2 |
15 | fib4 30983 | . . . 4 ⊢ (Fibci‘4) = 3 | |
16 | 14, 15 | oveq12i 6890 | . . 3 ⊢ ((Fibci‘(4 − 1)) + (Fibci‘4)) = (2 + 3) |
17 | 2cn 11388 | . . . 4 ⊢ 2 ∈ ℂ | |
18 | 3p2e5 11471 | . . . 4 ⊢ (3 + 2) = 5 | |
19 | 8, 17, 18 | addcomli 10518 | . . 3 ⊢ (2 + 3) = 5 |
20 | 5, 16, 19 | 3eqtri 2825 | . 2 ⊢ (Fibci‘(4 + 1)) = 5 |
21 | 2, 20 | eqtr3i 2823 | 1 ⊢ (Fibci‘5) = 5 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ‘cfv 6101 (class class class)co 6878 1c1 10225 + caddc 10227 − cmin 10556 ℕcn 11312 2c2 11368 3c3 11369 4c4 11370 5c5 11371 Fibcicfib 30975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-n0 11581 df-xnn0 11653 df-z 11667 df-uz 11931 df-rp 12075 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-word 13535 df-lsw 13583 df-concat 13591 df-s1 13616 df-substr 13665 df-pfx 13714 df-s2 13933 df-sseq 30962 df-fib 30976 |
This theorem is referenced by: fib6 30985 |
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