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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fib4 | Structured version Visualization version GIF version |
Description: Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
Ref | Expression |
---|---|
fib4 | ⊢ (Fibci‘4) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3p1e4 12409 | . . 3 ⊢ (3 + 1) = 4 | |
2 | 1 | fveq2i 6910 | . 2 ⊢ (Fibci‘(3 + 1)) = (Fibci‘4) |
3 | 3nn 12343 | . . . 4 ⊢ 3 ∈ ℕ | |
4 | fibp1 34383 | . . . 4 ⊢ (3 ∈ ℕ → (Fibci‘(3 + 1)) = ((Fibci‘(3 − 1)) + (Fibci‘3))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (Fibci‘(3 + 1)) = ((Fibci‘(3 − 1)) + (Fibci‘3)) |
6 | 3m1e2 12392 | . . . . . 6 ⊢ (3 − 1) = 2 | |
7 | 6 | fveq2i 6910 | . . . . 5 ⊢ (Fibci‘(3 − 1)) = (Fibci‘2) |
8 | fib2 34384 | . . . . 5 ⊢ (Fibci‘2) = 1 | |
9 | 7, 8 | eqtri 2763 | . . . 4 ⊢ (Fibci‘(3 − 1)) = 1 |
10 | fib3 34385 | . . . 4 ⊢ (Fibci‘3) = 2 | |
11 | 9, 10 | oveq12i 7443 | . . 3 ⊢ ((Fibci‘(3 − 1)) + (Fibci‘3)) = (1 + 2) |
12 | 1p2e3 12407 | . . 3 ⊢ (1 + 2) = 3 | |
13 | 5, 11, 12 | 3eqtri 2767 | . 2 ⊢ (Fibci‘(3 + 1)) = 3 |
14 | 2, 13 | eqtr3i 2765 | 1 ⊢ (Fibci‘4) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 − cmin 11490 ℕcn 12264 2c2 12319 3c3 12320 4c4 12321 Fibcicfib 34378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-s2 14884 df-sseq 34366 df-fib 34379 |
This theorem is referenced by: fib5 34387 fib6 34388 |
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