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Mirrors > Home > MPE Home > Th. List > prmo4 | Structured version Visualization version GIF version |
Description: The primorial of 4. (Contributed by AV, 28-Aug-2020.) |
Ref | Expression |
---|---|
prmo4 | ⊢ (#p‘4) = 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn 11573 | . . . 4 ⊢ 4 ∈ ℕ | |
2 | prmonn2 16209 | . . . 4 ⊢ (4 ∈ ℕ → (#p‘4) = if(4 ∈ ℙ, ((#p‘(4 − 1)) · 4), (#p‘(4 − 1)))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (#p‘4) = if(4 ∈ ℙ, ((#p‘(4 − 1)) · 4), (#p‘(4 − 1))) |
4 | 4nprm 15873 | . . . 4 ⊢ ¬ 4 ∈ ℙ | |
5 | 4 | iffalsei 4395 | . . 3 ⊢ if(4 ∈ ℙ, ((#p‘(4 − 1)) · 4), (#p‘(4 − 1))) = (#p‘(4 − 1)) |
6 | 3, 5 | eqtri 2819 | . 2 ⊢ (#p‘4) = (#p‘(4 − 1)) |
7 | 4m1e3 11619 | . . . 4 ⊢ (4 − 1) = 3 | |
8 | 7 | fveq2i 6546 | . . 3 ⊢ (#p‘(4 − 1)) = (#p‘3) |
9 | prmo3 16211 | . . 3 ⊢ (#p‘3) = 6 | |
10 | 8, 9 | eqtri 2819 | . 2 ⊢ (#p‘(4 − 1)) = 6 |
11 | 6, 10 | eqtri 2819 | 1 ⊢ (#p‘4) = 6 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 ifcif 4385 ‘cfv 6230 (class class class)co 7021 1c1 10389 · cmul 10393 − cmin 10722 ℕcn 11491 3c3 11546 4c4 11547 6c6 11549 ℙcprime 15849 #pcprmo 16201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-sup 8757 df-oi 8825 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-n0 11751 df-z 11835 df-uz 12099 df-rp 12245 df-fz 12748 df-fzo 12889 df-seq 13225 df-exp 13285 df-hash 13546 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-clim 14684 df-prod 15098 df-dvds 15446 df-prm 15850 df-prmo 16202 |
This theorem is referenced by: prmo5 16296 |
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