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 11784 | . . 3 ⊢ (4 + 1) = 5 | |
2 | 1 | fveq2i 6673 | . 2 ⊢ (Fibci‘(4 + 1)) = (Fibci‘5) |
3 | 4nn 11721 | . . . 4 ⊢ 4 ∈ ℕ | |
4 | fibp1 31659 | . . . 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 11723 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
7 | ax-1cn 10595 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 3cn 11719 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
9 | 3p1e4 11783 | . . . . . . . 8 ⊢ (3 + 1) = 4 | |
10 | 8, 7, 9 | addcomli 10832 | . . . . . . 7 ⊢ (1 + 3) = 4 |
11 | 6, 7, 8, 10 | subaddrii 10975 | . . . . . 6 ⊢ (4 − 1) = 3 |
12 | 11 | fveq2i 6673 | . . . . 5 ⊢ (Fibci‘(4 − 1)) = (Fibci‘3) |
13 | fib3 31661 | . . . . 5 ⊢ (Fibci‘3) = 2 | |
14 | 12, 13 | eqtri 2844 | . . . 4 ⊢ (Fibci‘(4 − 1)) = 2 |
15 | fib4 31662 | . . . 4 ⊢ (Fibci‘4) = 3 | |
16 | 14, 15 | oveq12i 7168 | . . 3 ⊢ ((Fibci‘(4 − 1)) + (Fibci‘4)) = (2 + 3) |
17 | 2cn 11713 | . . . 4 ⊢ 2 ∈ ℂ | |
18 | 3p2e5 11789 | . . . 4 ⊢ (3 + 2) = 5 | |
19 | 8, 17, 18 | addcomli 10832 | . . 3 ⊢ (2 + 3) = 5 |
20 | 5, 16, 19 | 3eqtri 2848 | . 2 ⊢ (Fibci‘(4 + 1)) = 5 |
21 | 2, 20 | eqtr3i 2846 | 1 ⊢ (Fibci‘5) = 5 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 1c1 10538 + caddc 10540 − cmin 10870 ℕcn 11638 2c2 11693 3c3 11694 4c4 11695 5c5 11696 Fibcicfib 31654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-word 13863 df-lsw 13915 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-s2 14210 df-sseq 31642 df-fib 31655 |
This theorem is referenced by: fib6 31664 |
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