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| Mirrors > Home > MPE Home > Th. List > ex-prmo | Structured version Visualization version GIF version | ||
| Description: Example for df-prmo 17070: (#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 12749 | . . . 4 ⊢ ;10 ∈ ℕ | |
| 2 | prmonn2 17077 | . . . 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 17151 | . . . 4 ⊢ ¬ ;10 ∈ ℙ | |
| 5 | 4 | iffalsei 4535 | . . 3 ⊢ if(;10 ∈ ℙ, ((#p‘(;10 − 1)) · ;10), (#p‘(;10 − 1))) = (#p‘(;10 − 1)) |
| 6 | 3, 5 | eqtri 2765 | . 2 ⊢ (#p‘;10) = (#p‘(;10 − 1)) |
| 7 | 10m1e9 12829 | . . 3 ⊢ (;10 − 1) = 9 | |
| 8 | 7 | fveq2i 6909 | . 2 ⊢ (#p‘(;10 − 1)) = (#p‘9) |
| 9 | 9nn 12364 | . . . . 5 ⊢ 9 ∈ ℕ | |
| 10 | prmonn2 17077 | . . . . 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 17150 | . . . . 5 ⊢ ¬ 9 ∈ ℙ | |
| 13 | 12 | iffalsei 4535 | . . . 4 ⊢ if(9 ∈ ℙ, ((#p‘(9 − 1)) · 9), (#p‘(9 − 1))) = (#p‘(9 − 1)) |
| 14 | 11, 13 | eqtri 2765 | . . 3 ⊢ (#p‘9) = (#p‘(9 − 1)) |
| 15 | 9m1e8 12400 | . . . 4 ⊢ (9 − 1) = 8 | |
| 16 | 15 | fveq2i 6909 | . . 3 ⊢ (#p‘(9 − 1)) = (#p‘8) |
| 17 | 8nn 12361 | . . . . . 6 ⊢ 8 ∈ ℕ | |
| 18 | prmonn2 17077 | . . . . . 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 17149 | . . . . . 6 ⊢ ¬ 8 ∈ ℙ | |
| 21 | 20 | iffalsei 4535 | . . . . 5 ⊢ if(8 ∈ ℙ, ((#p‘(8 − 1)) · 8), (#p‘(8 − 1))) = (#p‘(8 − 1)) |
| 22 | 19, 21 | eqtri 2765 | . . . 4 ⊢ (#p‘8) = (#p‘(8 − 1)) |
| 23 | 8m1e7 12399 | . . . . 5 ⊢ (8 − 1) = 7 | |
| 24 | 23 | fveq2i 6909 | . . . 4 ⊢ (#p‘(8 − 1)) = (#p‘7) |
| 25 | 7nn 12358 | . . . . . 6 ⊢ 7 ∈ ℕ | |
| 26 | prmonn2 17077 | . . . . . 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 17148 | . . . . . 6 ⊢ 7 ∈ ℙ | |
| 29 | 28 | iftruei 4532 | . . . . 5 ⊢ if(7 ∈ ℙ, ((#p‘(7 − 1)) · 7), (#p‘(7 − 1))) = ((#p‘(7 − 1)) · 7) |
| 30 | 7nn0 12548 | . . . . . 6 ⊢ 7 ∈ ℕ0 | |
| 31 | 3nn0 12544 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 32 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 33 | 7m1e6 12398 | . . . . . . . 8 ⊢ (7 − 1) = 6 | |
| 34 | 33 | fveq2i 6909 | . . . . . . 7 ⊢ (#p‘(7 − 1)) = (#p‘6) |
| 35 | prmo6 17167 | . . . . . . 7 ⊢ (#p‘6) = ;30 | |
| 36 | 34, 35 | eqtri 2765 | . . . . . 6 ⊢ (#p‘(7 − 1)) = ;30 |
| 37 | 7cn 12360 | . . . . . . 7 ⊢ 7 ∈ ℂ | |
| 38 | 3cn 12347 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 39 | 7t3e21 12843 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
| 40 | 37, 38, 39 | mulcomli 11270 | . . . . . 6 ⊢ (3 · 7) = ;21 |
| 41 | 37 | mul02i 11450 | . . . . . 6 ⊢ (0 · 7) = 0 |
| 42 | 30, 31, 32, 36, 40, 41 | decmul1 12797 | . . . . 5 ⊢ ((#p‘(7 − 1)) · 7) = ;;210 |
| 43 | 27, 29, 42 | 3eqtri 2769 | . . . 4 ⊢ (#p‘7) = ;;210 |
| 44 | 22, 24, 43 | 3eqtri 2769 | . . 3 ⊢ (#p‘8) = ;;210 |
| 45 | 14, 16, 44 | 3eqtri 2769 | . 2 ⊢ (#p‘9) = ;;210 |
| 46 | 6, 8, 45 | 3eqtri 2769 | 1 ⊢ (#p‘;10) = ;;210 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ifcif 4525 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 − cmin 11492 ℕcn 12266 2c2 12321 3c3 12322 6c6 12325 7c7 12326 8c8 12327 9c9 12328 ;cdc 12733 ℙcprime 16708 #pcprmo 17069 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-dvds 16291 df-prm 16709 df-prmo 17070 |
| This theorem is referenced by: (None) |
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