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Theorem prmop1 15742
Description: The primorial of a successor. (Contributed by AV, 28-Aug-2020.)
Assertion
Ref Expression
prmop1 |- ( N e. NN0 -> (#p ` ( N + 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) )
Proof of Theorem prmop1
Dummy variable k is distinct from all other variables.
StepHypRef Expression
1 peano2nn0 11333 . . 3 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
2 prmoval 15737 . . 3 |- ( ( N + 1 ) e. NN0 -> (#p ` ( N + 1 ) ) = prod_ k e. ( 1 ... ( N + 1 ) ) if ( k e. Prime , k , 1 ) )
31, 2syl 17 . 2 |- ( N e. NN0 -> (#p ` ( N + 1 ) ) = prod_ k e. ( 1 ... ( N + 1 ) ) if ( k e. Prime , k , 1 ) )
4 nn0p1nn 11332 . . . 4 |- ( N e. NN0 -> ( N + 1 ) e. NN )
5 elnnuz 11724 . . . 4 |- ( ( N + 1 ) e. NN <-> ( N + 1 ) e. ( ZZ>= ` 1 ) )
64, 5sylib 208 . . 3 |- ( N e. NN0 -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
7 elfzelz 12342 . . . . . 6 |- ( k e. ( 1 ... ( N + 1 ) ) -> k e. ZZ )
87zcnd 11483 . . . . 5 |- ( k e. ( 1 ... ( N + 1 ) ) -> k e. CC )
98adantl 482 . . . 4 |- ( ( N e. NN0 /\ k e. ( 1 ... ( N + 1 ) ) ) -> k e. CC )
10 1cnd 10056 . . . 4 |- ( ( N e. NN0 /\ k e. ( 1 ... ( N + 1 ) ) ) -> 1 e. CC )
119, 10ifcld 4131 . . 3 |- ( ( N e. NN0 /\ k e. ( 1 ... ( N + 1 ) ) ) -> if ( k e. Prime , k , 1 ) e. CC )
12 eleq1 2689 . . . 4 |- ( k = ( N + 1 ) -> ( k e. Prime <-> ( N + 1 ) e. Prime ) )
13 id 22 . . . 4 |- ( k = ( N + 1 ) -> k = ( N + 1 ) )
1412, 13ifbieq1d 4109 . . 3 |- ( k = ( N + 1 ) -> if ( k e. Prime , k , 1 ) = if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) )
156, 11, 14fprodm1 14697 . 2 |- ( N e. NN0 -> prod_ k e. ( 1 ... ( N + 1 ) ) if ( k e. Prime , k , 1 ) = ( prod_ k e. ( 1 ... ( ( N + 1 ) - 1 ) ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) )
16 nn0cn 11302 . . . . . . 7 |- ( N e. NN0 -> N e. CC )
17 pncan1 10454 . . . . . . 7 |- ( N e. CC -> ( ( N + 1 ) - 1 ) = N )
1816, 17syl 17 . . . . . 6 |- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N )
1918oveq2d 6666 . . . . 5 |- ( N e. NN0 -> ( 1 ... ( ( N + 1 ) - 1 ) ) = ( 1 ... N ) )
2019prodeq1d 14651 . . . 4 |- ( N e. NN0 -> prod_ k e. ( 1 ... ( ( N + 1 ) - 1 ) ) if ( k e. Prime , k , 1 ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
2120oveq1d 6665 . . 3 |- ( N e. NN0 -> ( prod_ k e. ( 1 ... ( ( N + 1 ) - 1 ) ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) )
22 prmoval 15737 . . . . . . . 8 |- ( N e. NN0 -> (#p ` N ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
2322eqcomd 2628 . . . . . . 7 |- ( N e. NN0 -> prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) = (#p ` N ) )
2423adantl 482 . . . . . 6 |- ( ( ( N + 1 ) e. Prime /\ N e. NN0 ) -> prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) = (#p ` N ) )
2524oveq1d 6665 . . . . 5 |- ( ( ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. ( N + 1 ) ) = ( (#p ` N ) x. ( N + 1 ) ) )
26 iftrue 4092 . . . . . . . 8 |- ( ( N + 1 ) e. Prime -> if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) = ( N + 1 ) )
2726oveq2d 6666 . . . . . . 7 |- ( ( N + 1 ) e. Prime -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. ( N + 1 ) ) )
28 iftrue 4092 . . . . . . 7 |- ( ( N + 1 ) e. Prime -> if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) = ( (#p ` N ) x. ( N + 1 ) ) )
2927, 28eqeq12d 2637 . . . . . 6 |- ( ( N + 1 ) e. Prime -> ( ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) <-> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. ( N + 1 ) ) = ( (#p ` N ) x. ( N + 1 ) ) ) )
3029adantr 481 . . . . 5 |- ( ( ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) <-> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. ( N + 1 ) ) = ( (#p ` N ) x. ( N + 1 ) ) ) )
3125, 30mpbird 247 . . . 4 |- ( ( ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) )
32 fzfid 12772 . . . . . . . . . 10 |- ( N e. NN0 -> ( 1 ... N ) e. Fin )
33 elfznn 12370 . . . . . . . . . . . 12 |- ( k e. ( 1 ... N ) -> k e. NN )
34 1nn 11031 . . . . . . . . . . . . 13 |- 1 e. NN
3534a1i 11 . . . . . . . . . . . 12 |- ( k e. ( 1 ... N ) -> 1 e. NN )
3633, 35ifcld 4131 . . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> if ( k e. Prime , k , 1 ) e. NN )
3736adantl 482 . . . . . . . . . 10 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> if ( k e. Prime , k , 1 ) e. NN )
3832, 37fprodnncl 14685 . . . . . . . . 9 |- ( N e. NN0 -> prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) e. NN )
3938nncnd 11036 . . . . . . . 8 |- ( N e. NN0 -> prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) e. CC )
4039adantl 482 . . . . . . 7 |- ( ( -. ( N + 1 ) e. Prime /\ N e. NN0 ) -> prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) e. CC )
4140mulid1d 10057 . . . . . 6 |- ( ( -. ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. 1 ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
4222adantl 482 . . . . . 6 |- ( ( -. ( N + 1 ) e. Prime /\ N e. NN0 ) -> (#p ` N ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
4341, 42eqtr4d 2659 . . . . 5 |- ( ( -. ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. 1 ) = (#p ` N ) )
44 iffalse 4095 . . . . . . . 8 |- ( -. ( N + 1 ) e. Prime -> if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) = 1 )
4544oveq2d 6666 . . . . . . 7 |- ( -. ( N + 1 ) e. Prime -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. 1 ) )
46 iffalse 4095 . . . . . . 7 |- ( -. ( N + 1 ) e. Prime -> if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) = (#p ` N ) )
4745, 46eqeq12d 2637 . . . . . 6 |- ( -. ( N + 1 ) e. Prime -> ( ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) <-> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. 1 ) = (#p ` N ) ) )
4847adantr 481 . . . . 5 |- ( ( -. ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) <-> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. 1 ) = (#p ` N ) ) )
4943, 48mpbird 247 . . . 4 |- ( ( -. ( N + 1 ) e. Prime /\ N e. NN0 ) -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) )
5031, 49pm2.61ian 831 . . 3 |- ( N e. NN0 -> ( prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) )
5121, 50eqtrd 2656 . 2 |- ( N e. NN0 -> ( prod_ k e. ( 1 ... ( ( N + 1 ) - 1 ) ) if ( k e. Prime , k , 1 ) x. if ( ( N + 1 ) e. Prime , ( N + 1 ) , 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) )
523, 15, 513eqtrd 2660 1 |- ( N e. NN0 -> (#p ` ( N + 1 ) ) = if ( ( N + 1 ) e. Prime , ( (#p ` N ) x. ( N + 1 ) ) , (#p ` N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   prod_cprod 14635   Primecprime 15385  #pcprmo 15735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-prmo 15736
This theorem is referenced by:  prmonn2  15743
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