Mathbox for Thierry Arnoux |
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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 11942 | . . 3 ⊢ (5 + 1) = 6 | |
2 | 1 | fveq2i 6698 | . 2 ⊢ (Fibci‘(5 + 1)) = (Fibci‘6) |
3 | 5nn 11881 | . . . 4 ⊢ 5 ∈ ℕ | |
4 | fibp1 32034 | . . . 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 11883 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
7 | ax-1cn 10752 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 4cn 11880 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
9 | 4p1e5 11941 | . . . . . . . 8 ⊢ (4 + 1) = 5 | |
10 | 8, 7, 9 | addcomli 10989 | . . . . . . 7 ⊢ (1 + 4) = 5 |
11 | 6, 7, 8, 10 | subaddrii 11132 | . . . . . 6 ⊢ (5 − 1) = 4 |
12 | 11 | fveq2i 6698 | . . . . 5 ⊢ (Fibci‘(5 − 1)) = (Fibci‘4) |
13 | fib4 32037 | . . . . 5 ⊢ (Fibci‘4) = 3 | |
14 | 12, 13 | eqtri 2759 | . . . 4 ⊢ (Fibci‘(5 − 1)) = 3 |
15 | fib5 32038 | . . . 4 ⊢ (Fibci‘5) = 5 | |
16 | 14, 15 | oveq12i 7203 | . . 3 ⊢ ((Fibci‘(5 − 1)) + (Fibci‘5)) = (3 + 5) |
17 | 3cn 11876 | . . . 4 ⊢ 3 ∈ ℂ | |
18 | 5p3e8 11952 | . . . 4 ⊢ (5 + 3) = 8 | |
19 | 6, 17, 18 | addcomli 10989 | . . 3 ⊢ (3 + 5) = 8 |
20 | 5, 16, 19 | 3eqtri 2763 | . 2 ⊢ (Fibci‘(5 + 1)) = 8 |
21 | 2, 20 | eqtr3i 2761 | 1 ⊢ (Fibci‘6) = 8 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 1c1 10695 + caddc 10697 − cmin 11027 ℕcn 11795 3c3 11851 4c4 11852 5c5 11853 6c6 11854 8c8 11856 Fibcicfib 32029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-rp 12552 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-word 14035 df-lsw 14083 df-concat 14091 df-s1 14118 df-substr 14171 df-pfx 14201 df-s2 14378 df-sseq 32017 df-fib 32030 |
This theorem is referenced by: (None) |
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