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 11772 | . . 3 ⊢ (5 + 1) = 6 | |
2 | 1 | fveq2i 6666 | . 2 ⊢ (Fibci‘(5 + 1)) = (Fibci‘6) |
3 | 5nn 11711 | . . . 4 ⊢ 5 ∈ ℕ | |
4 | fibp1 31558 | . . . 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 11713 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
7 | ax-1cn 10583 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 4cn 11710 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
9 | 4p1e5 11771 | . . . . . . . 8 ⊢ (4 + 1) = 5 | |
10 | 8, 7, 9 | addcomli 10820 | . . . . . . 7 ⊢ (1 + 4) = 5 |
11 | 6, 7, 8, 10 | subaddrii 10963 | . . . . . 6 ⊢ (5 − 1) = 4 |
12 | 11 | fveq2i 6666 | . . . . 5 ⊢ (Fibci‘(5 − 1)) = (Fibci‘4) |
13 | fib4 31561 | . . . . 5 ⊢ (Fibci‘4) = 3 | |
14 | 12, 13 | eqtri 2841 | . . . 4 ⊢ (Fibci‘(5 − 1)) = 3 |
15 | fib5 31562 | . . . 4 ⊢ (Fibci‘5) = 5 | |
16 | 14, 15 | oveq12i 7157 | . . 3 ⊢ ((Fibci‘(5 − 1)) + (Fibci‘5)) = (3 + 5) |
17 | 3cn 11706 | . . . 4 ⊢ 3 ∈ ℂ | |
18 | 5p3e8 11782 | . . . 4 ⊢ (5 + 3) = 8 | |
19 | 6, 17, 18 | addcomli 10820 | . . 3 ⊢ (3 + 5) = 8 |
20 | 5, 16, 19 | 3eqtri 2845 | . 2 ⊢ (Fibci‘(5 + 1)) = 8 |
21 | 2, 20 | eqtr3i 2843 | 1 ⊢ (Fibci‘6) = 8 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 1c1 10526 + caddc 10528 − cmin 10858 ℕcn 11626 3c3 11681 4c4 11682 5c5 11683 6c6 11684 8c8 11686 Fibcicfib 31553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-word 13850 df-lsw 13903 df-concat 13911 df-s1 13938 df-substr 13991 df-pfx 14021 df-s2 14198 df-sseq 31541 df-fib 31554 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |