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Mirrors > Home > MPE Home > Th. List > ex-prmo | Structured version Visualization version GIF version |
Description: Example for df-prmo 16731: (#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 12452 | . . . 4 ⊢ ;10 ∈ ℕ | |
2 | prmonn2 16738 | . . . 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 16813 | . . . 4 ⊢ ¬ ;10 ∈ ℙ | |
5 | 4 | iffalsei 4475 | . . 3 ⊢ if(;10 ∈ ℙ, ((#p‘(;10 − 1)) · ;10), (#p‘(;10 − 1))) = (#p‘(;10 − 1)) |
6 | 3, 5 | eqtri 2768 | . 2 ⊢ (#p‘;10) = (#p‘(;10 − 1)) |
7 | 10m1e9 12532 | . . 3 ⊢ (;10 − 1) = 9 | |
8 | 7 | fveq2i 6774 | . 2 ⊢ (#p‘(;10 − 1)) = (#p‘9) |
9 | 9nn 12071 | . . . . 5 ⊢ 9 ∈ ℕ | |
10 | prmonn2 16738 | . . . . 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 16812 | . . . . 5 ⊢ ¬ 9 ∈ ℙ | |
13 | 12 | iffalsei 4475 | . . . 4 ⊢ if(9 ∈ ℙ, ((#p‘(9 − 1)) · 9), (#p‘(9 − 1))) = (#p‘(9 − 1)) |
14 | 11, 13 | eqtri 2768 | . . 3 ⊢ (#p‘9) = (#p‘(9 − 1)) |
15 | 9m1e8 12107 | . . . 4 ⊢ (9 − 1) = 8 | |
16 | 15 | fveq2i 6774 | . . 3 ⊢ (#p‘(9 − 1)) = (#p‘8) |
17 | 8nn 12068 | . . . . . 6 ⊢ 8 ∈ ℕ | |
18 | prmonn2 16738 | . . . . . 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 16811 | . . . . . 6 ⊢ ¬ 8 ∈ ℙ | |
21 | 20 | iffalsei 4475 | . . . . 5 ⊢ if(8 ∈ ℙ, ((#p‘(8 − 1)) · 8), (#p‘(8 − 1))) = (#p‘(8 − 1)) |
22 | 19, 21 | eqtri 2768 | . . . 4 ⊢ (#p‘8) = (#p‘(8 − 1)) |
23 | 8m1e7 12106 | . . . . 5 ⊢ (8 − 1) = 7 | |
24 | 23 | fveq2i 6774 | . . . 4 ⊢ (#p‘(8 − 1)) = (#p‘7) |
25 | 7nn 12065 | . . . . . 6 ⊢ 7 ∈ ℕ | |
26 | prmonn2 16738 | . . . . . 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 16810 | . . . . . 6 ⊢ 7 ∈ ℙ | |
29 | 28 | iftruei 4472 | . . . . 5 ⊢ if(7 ∈ ℙ, ((#p‘(7 − 1)) · 7), (#p‘(7 − 1))) = ((#p‘(7 − 1)) · 7) |
30 | 7nn0 12255 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
31 | 3nn0 12251 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
32 | 0nn0 12248 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
33 | 7m1e6 12105 | . . . . . . . 8 ⊢ (7 − 1) = 6 | |
34 | 33 | fveq2i 6774 | . . . . . . 7 ⊢ (#p‘(7 − 1)) = (#p‘6) |
35 | prmo6 16829 | . . . . . . 7 ⊢ (#p‘6) = ;30 | |
36 | 34, 35 | eqtri 2768 | . . . . . 6 ⊢ (#p‘(7 − 1)) = ;30 |
37 | 7cn 12067 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
38 | 3cn 12054 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
39 | 7t3e21 12546 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
40 | 37, 38, 39 | mulcomli 10985 | . . . . . 6 ⊢ (3 · 7) = ;21 |
41 | 37 | mul02i 11164 | . . . . . 6 ⊢ (0 · 7) = 0 |
42 | 30, 31, 32, 36, 40, 41 | decmul1 12500 | . . . . 5 ⊢ ((#p‘(7 − 1)) · 7) = ;;210 |
43 | 27, 29, 42 | 3eqtri 2772 | . . . 4 ⊢ (#p‘7) = ;;210 |
44 | 22, 24, 43 | 3eqtri 2772 | . . 3 ⊢ (#p‘8) = ;;210 |
45 | 14, 16, 44 | 3eqtri 2772 | . 2 ⊢ (#p‘9) = ;;210 |
46 | 6, 8, 45 | 3eqtri 2772 | 1 ⊢ (#p‘;10) = ;;210 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ifcif 4465 ‘cfv 6432 (class class class)co 7271 0cc0 10872 1c1 10873 · cmul 10877 − cmin 11205 ℕcn 11973 2c2 12028 3c3 12029 6c6 12032 7c7 12033 8c8 12034 9c9 12035 ;cdc 12436 ℙcprime 16374 #pcprmo 16730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-prod 15614 df-dvds 15962 df-prm 16375 df-prmo 16731 |
This theorem is referenced by: (None) |
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