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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 12325 | . . . 4 ⊢ 4 ∈ ℕ | |
2 | prmonn2 17007 | . . . 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 16665 | . . . 4 ⊢ ¬ 4 ∈ ℙ | |
5 | 4 | iffalsei 4539 | . . 3 ⊢ if(4 ∈ ℙ, ((#p‘(4 − 1)) · 4), (#p‘(4 − 1))) = (#p‘(4 − 1)) |
6 | 3, 5 | eqtri 2756 | . 2 ⊢ (#p‘4) = (#p‘(4 − 1)) |
7 | 4m1e3 12371 | . . . 4 ⊢ (4 − 1) = 3 | |
8 | 7 | fveq2i 6900 | . . 3 ⊢ (#p‘(4 − 1)) = (#p‘3) |
9 | prmo3 17009 | . . 3 ⊢ (#p‘3) = 6 | |
10 | 8, 9 | eqtri 2756 | . 2 ⊢ (#p‘(4 − 1)) = 6 |
11 | 6, 10 | eqtri 2756 | 1 ⊢ (#p‘4) = 6 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ifcif 4529 ‘cfv 6548 (class class class)co 7420 1c1 11139 · cmul 11143 − cmin 11474 ℕcn 12242 3c3 12298 4c4 12299 6c6 12301 ℙcprime 16641 #pcprmo 16999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-prod 15882 df-dvds 16231 df-prm 16642 df-prmo 17000 |
This theorem is referenced by: prmo5 17097 |
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