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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 12354 | . . 3 ⊢ (4 + 1) = 5 | |
2 | 1 | fveq2i 6891 | . 2 ⊢ (Fibci‘(4 + 1)) = (Fibci‘5) |
3 | 4nn 12291 | . . . 4 ⊢ 4 ∈ ℕ | |
4 | fibp1 33338 | . . . 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 12293 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
7 | ax-1cn 11164 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 3cn 12289 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
9 | 3p1e4 12353 | . . . . . . . 8 ⊢ (3 + 1) = 4 | |
10 | 8, 7, 9 | addcomli 11402 | . . . . . . 7 ⊢ (1 + 3) = 4 |
11 | 6, 7, 8, 10 | subaddrii 11545 | . . . . . 6 ⊢ (4 − 1) = 3 |
12 | 11 | fveq2i 6891 | . . . . 5 ⊢ (Fibci‘(4 − 1)) = (Fibci‘3) |
13 | fib3 33340 | . . . . 5 ⊢ (Fibci‘3) = 2 | |
14 | 12, 13 | eqtri 2761 | . . . 4 ⊢ (Fibci‘(4 − 1)) = 2 |
15 | fib4 33341 | . . . 4 ⊢ (Fibci‘4) = 3 | |
16 | 14, 15 | oveq12i 7416 | . . 3 ⊢ ((Fibci‘(4 − 1)) + (Fibci‘4)) = (2 + 3) |
17 | 2cn 12283 | . . . 4 ⊢ 2 ∈ ℂ | |
18 | 3p2e5 12359 | . . . 4 ⊢ (3 + 2) = 5 | |
19 | 8, 17, 18 | addcomli 11402 | . . 3 ⊢ (2 + 3) = 5 |
20 | 5, 16, 19 | 3eqtri 2765 | . 2 ⊢ (Fibci‘(4 + 1)) = 5 |
21 | 2, 20 | eqtr3i 2763 | 1 ⊢ (Fibci‘5) = 5 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 1c1 11107 + caddc 11109 − cmin 11440 ℕcn 12208 2c2 12263 3c3 12264 4c4 12265 5c5 12266 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-3 12272 df-4 12273 df-5 12274 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-substr 14587 df-pfx 14617 df-s2 14795 df-sseq 33321 df-fib 33334 |
This theorem is referenced by: fib6 33343 |
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