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Mirrors > Home > MPE Home > Th. List > ex-prmo | Structured version Visualization version GIF version |
Description: Example for df-prmo 15938: (#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 11706 | . . . 4 ⊢ ;10 ∈ ℕ | |
2 | prmonn2 15945 | . . . 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 16022 | . . . 4 ⊢ ¬ ;10 ∈ ℙ | |
5 | 4 | iffalsei 4240 | . . 3 ⊢ if(;10 ∈ ℙ, ((#p‘(;10 − 1)) · ;10), (#p‘(;10 − 1))) = (#p‘(;10 − 1)) |
6 | 3, 5 | eqtri 2782 | . 2 ⊢ (#p‘;10) = (#p‘(;10 − 1)) |
7 | 10m1e9 11822 | . . 3 ⊢ (;10 − 1) = 9 | |
8 | 7 | fveq2i 6355 | . 2 ⊢ (#p‘(;10 − 1)) = (#p‘9) |
9 | 9nn 11384 | . . . . 5 ⊢ 9 ∈ ℕ | |
10 | prmonn2 15945 | . . . . 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 16021 | . . . . 5 ⊢ ¬ 9 ∈ ℙ | |
13 | 12 | iffalsei 4240 | . . . 4 ⊢ if(9 ∈ ℙ, ((#p‘(9 − 1)) · 9), (#p‘(9 − 1))) = (#p‘(9 − 1)) |
14 | 11, 13 | eqtri 2782 | . . 3 ⊢ (#p‘9) = (#p‘(9 − 1)) |
15 | 9m1e8 11335 | . . . 4 ⊢ (9 − 1) = 8 | |
16 | 15 | fveq2i 6355 | . . 3 ⊢ (#p‘(9 − 1)) = (#p‘8) |
17 | 8nn 11383 | . . . . . 6 ⊢ 8 ∈ ℕ | |
18 | prmonn2 15945 | . . . . . 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 16020 | . . . . . 6 ⊢ ¬ 8 ∈ ℙ | |
21 | 20 | iffalsei 4240 | . . . . 5 ⊢ if(8 ∈ ℙ, ((#p‘(8 − 1)) · 8), (#p‘(8 − 1))) = (#p‘(8 − 1)) |
22 | 19, 21 | eqtri 2782 | . . . 4 ⊢ (#p‘8) = (#p‘(8 − 1)) |
23 | 8m1e7 11334 | . . . . 5 ⊢ (8 − 1) = 7 | |
24 | 23 | fveq2i 6355 | . . . 4 ⊢ (#p‘(8 − 1)) = (#p‘7) |
25 | 7nn 11382 | . . . . . 6 ⊢ 7 ∈ ℕ | |
26 | prmonn2 15945 | . . . . . 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 16019 | . . . . . 6 ⊢ 7 ∈ ℙ | |
29 | 28 | iftruei 4237 | . . . . 5 ⊢ if(7 ∈ ℙ, ((#p‘(7 − 1)) · 7), (#p‘(7 − 1))) = ((#p‘(7 − 1)) · 7) |
30 | 7nn0 11506 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
31 | 3nn0 11502 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
32 | 0nn0 11499 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
33 | 7m1e6 11333 | . . . . . . . 8 ⊢ (7 − 1) = 6 | |
34 | 33 | fveq2i 6355 | . . . . . . 7 ⊢ (#p‘(7 − 1)) = (#p‘6) |
35 | prmo6 16039 | . . . . . . 7 ⊢ (#p‘6) = ;30 | |
36 | 34, 35 | eqtri 2782 | . . . . . 6 ⊢ (#p‘(7 − 1)) = ;30 |
37 | 7cn 11296 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
38 | 3cn 11287 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
39 | 7t3e21 11841 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
40 | 37, 38, 39 | mulcomli 10239 | . . . . . 6 ⊢ (3 · 7) = ;21 |
41 | 37 | mul02i 10417 | . . . . . 6 ⊢ (0 · 7) = 0 |
42 | 30, 31, 32, 36, 32, 40, 41 | decmul1 11777 | . . . . 5 ⊢ ((#p‘(7 − 1)) · 7) = ;;210 |
43 | 27, 29, 42 | 3eqtri 2786 | . . . 4 ⊢ (#p‘7) = ;;210 |
44 | 22, 24, 43 | 3eqtri 2786 | . . 3 ⊢ (#p‘8) = ;;210 |
45 | 14, 16, 44 | 3eqtri 2786 | . 2 ⊢ (#p‘9) = ;;210 |
46 | 6, 8, 45 | 3eqtri 2786 | 1 ⊢ (#p‘;10) = ;;210 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 ifcif 4230 ‘cfv 6049 (class class class)co 6813 0cc0 10128 1c1 10129 · cmul 10133 − cmin 10458 ℕcn 11212 2c2 11262 3c3 11263 6c6 11266 7c7 11267 8c8 11268 9c9 11269 ;cdc 11685 ℙcprime 15587 #pcprmo 15937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-rp 12026 df-fz 12520 df-fzo 12660 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-prod 14835 df-dvds 15183 df-prm 15588 df-prmo 15938 |
This theorem is referenced by: (None) |
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