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Mirrors > Home > MPE Home > Th. List > ex-prmo | Structured version Visualization version GIF version |
Description: Example for df-prmo 16114: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
ex-prmo | ⊢ (#p‘;10) = ;;210 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn 11844 | . . . 4 ⊢ ;10 ∈ ℕ | |
2 | prmonn2 16121 | . . . 4 ⊢ (;10 ∈ ℕ → (#p‘;10) = if(;10 ∈ ℙ, ((#p‘(;10 − 1)) · ;10), (#p‘(;10 − 1)))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (#p‘;10) = if(;10 ∈ ℙ, ((#p‘(;10 − 1)) · ;10), (#p‘(;10 − 1))) |
4 | 10nprm 16193 | . . . 4 ⊢ ¬ ;10 ∈ ℙ | |
5 | 4 | iffalsei 4318 | . . 3 ⊢ if(;10 ∈ ℙ, ((#p‘(;10 − 1)) · ;10), (#p‘(;10 − 1))) = (#p‘(;10 − 1)) |
6 | 3, 5 | eqtri 2849 | . 2 ⊢ (#p‘;10) = (#p‘(;10 − 1)) |
7 | 10m1e9 11926 | . . 3 ⊢ (;10 − 1) = 9 | |
8 | 7 | fveq2i 6440 | . 2 ⊢ (#p‘(;10 − 1)) = (#p‘9) |
9 | 9nn 11462 | . . . . 5 ⊢ 9 ∈ ℕ | |
10 | prmonn2 16121 | . . . . 5 ⊢ (9 ∈ ℕ → (#p‘9) = if(9 ∈ ℙ, ((#p‘(9 − 1)) · 9), (#p‘(9 − 1)))) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ (#p‘9) = if(9 ∈ ℙ, ((#p‘(9 − 1)) · 9), (#p‘(9 − 1))) |
12 | 9nprm 16192 | . . . . 5 ⊢ ¬ 9 ∈ ℙ | |
13 | 12 | iffalsei 4318 | . . . 4 ⊢ if(9 ∈ ℙ, ((#p‘(9 − 1)) · 9), (#p‘(9 − 1))) = (#p‘(9 − 1)) |
14 | 11, 13 | eqtri 2849 | . . 3 ⊢ (#p‘9) = (#p‘(9 − 1)) |
15 | 9m1e8 11499 | . . . 4 ⊢ (9 − 1) = 8 | |
16 | 15 | fveq2i 6440 | . . 3 ⊢ (#p‘(9 − 1)) = (#p‘8) |
17 | 8nn 11458 | . . . . . 6 ⊢ 8 ∈ ℕ | |
18 | prmonn2 16121 | . . . . . 6 ⊢ (8 ∈ ℕ → (#p‘8) = if(8 ∈ ℙ, ((#p‘(8 − 1)) · 8), (#p‘(8 − 1)))) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ (#p‘8) = if(8 ∈ ℙ, ((#p‘(8 − 1)) · 8), (#p‘(8 − 1))) |
20 | 8nprm 16191 | . . . . . 6 ⊢ ¬ 8 ∈ ℙ | |
21 | 20 | iffalsei 4318 | . . . . 5 ⊢ if(8 ∈ ℙ, ((#p‘(8 − 1)) · 8), (#p‘(8 − 1))) = (#p‘(8 − 1)) |
22 | 19, 21 | eqtri 2849 | . . . 4 ⊢ (#p‘8) = (#p‘(8 − 1)) |
23 | 8m1e7 11498 | . . . . 5 ⊢ (8 − 1) = 7 | |
24 | 23 | fveq2i 6440 | . . . 4 ⊢ (#p‘(8 − 1)) = (#p‘7) |
25 | 7nn 11454 | . . . . . 6 ⊢ 7 ∈ ℕ | |
26 | prmonn2 16121 | . . . . . 6 ⊢ (7 ∈ ℕ → (#p‘7) = if(7 ∈ ℙ, ((#p‘(7 − 1)) · 7), (#p‘(7 − 1)))) | |
27 | 25, 26 | ax-mp 5 | . . . . 5 ⊢ (#p‘7) = if(7 ∈ ℙ, ((#p‘(7 − 1)) · 7), (#p‘(7 − 1))) |
28 | 7prm 16190 | . . . . . 6 ⊢ 7 ∈ ℙ | |
29 | 28 | iftruei 4315 | . . . . 5 ⊢ if(7 ∈ ℙ, ((#p‘(7 − 1)) · 7), (#p‘(7 − 1))) = ((#p‘(7 − 1)) · 7) |
30 | 7nn0 11649 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
31 | 3nn0 11645 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
32 | 0nn0 11642 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
33 | 7m1e6 11497 | . . . . . . . 8 ⊢ (7 − 1) = 6 | |
34 | 33 | fveq2i 6440 | . . . . . . 7 ⊢ (#p‘(7 − 1)) = (#p‘6) |
35 | prmo6 16209 | . . . . . . 7 ⊢ (#p‘6) = ;30 | |
36 | 34, 35 | eqtri 2849 | . . . . . 6 ⊢ (#p‘(7 − 1)) = ;30 |
37 | 7cn 11456 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
38 | 3cn 11439 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
39 | 7t3e21 11940 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
40 | 37, 38, 39 | mulcomli 10373 | . . . . . 6 ⊢ (3 · 7) = ;21 |
41 | 37 | mul02i 10551 | . . . . . 6 ⊢ (0 · 7) = 0 |
42 | 30, 31, 32, 36, 40, 41 | decmul1 11893 | . . . . 5 ⊢ ((#p‘(7 − 1)) · 7) = ;;210 |
43 | 27, 29, 42 | 3eqtri 2853 | . . . 4 ⊢ (#p‘7) = ;;210 |
44 | 22, 24, 43 | 3eqtri 2853 | . . 3 ⊢ (#p‘8) = ;;210 |
45 | 14, 16, 44 | 3eqtri 2853 | . 2 ⊢ (#p‘9) = ;;210 |
46 | 6, 8, 45 | 3eqtri 2853 | 1 ⊢ (#p‘;10) = ;;210 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 ifcif 4308 ‘cfv 6127 (class class class)co 6910 0cc0 10259 1c1 10260 · cmul 10264 − cmin 10592 ℕcn 11357 2c2 11413 3c3 11414 6c6 11417 7c7 11418 8c8 11419 9c9 11420 ;cdc 11828 ℙcprime 15764 #pcprmo 16113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-prod 15016 df-dvds 15365 df-prm 15765 df-prmo 16114 |
This theorem is referenced by: (None) |
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