Theorem dvdsprmpweqnn 12289

Description: If an integer greater than 1 divides a prime power, it is a (proper) prime power. (Contributed by AV, 13-Aug-2021.)

Assertion

Ref	Expression
dvdsprmpweqnn	|- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) )

Distinct variable groups:   A, n   n, N   P, n

Proof of Theorem dvdsprmpweqnn

Step	Hyp	Ref	Expression
1		eluz2nn 9525	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
2		dvdsprmpweq 12288	. . . . 5 |- ( ( P e. Prime /\ A e. NN /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
3	1, 2	syl3an2 1267	. . . 4 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
4	3	imp 123	. . 3 |- ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN0 A = ( P ^ n ) )
5		df-n0 9136	. . . . . 6 |- NN0 = ( NN u. { 0 } )
6	5	rexeqi 2670	. . . . 5 |- ( E. n e. NN0 A = ( P ^ n ) <-> E. n e. ( NN u. { 0 } ) A = ( P ^ n ) )
7		rexun 3307	. . . . 5 |- ( E. n e. ( NN u. { 0 } ) A = ( P ^ n ) <-> ( E. n e. NN A = ( P ^ n ) \/ E. n e. { 0 } A = ( P ^ n ) ) )
8	6, 7	bitri 183	. . . 4 |- ( E. n e. NN0 A = ( P ^ n ) <-> ( E. n e. NN A = ( P ^ n ) \/ E. n e. { 0 } A = ( P ^ n ) ) )
9		0z 9223	. . . . . . 7 |- 0 e. ZZ
10		oveq2 5861	. . . . . . . . 9 |- ( n = 0 -> ( P ^ n ) = ( P ^ 0 ) )
11	10	eqeq2d 2182	. . . . . . . 8 |- ( n = 0 -> ( A = ( P ^ n ) <-> A = ( P ^ 0 ) ) )
12	11	rexsng 3624	. . . . . . 7 |- ( 0 e. ZZ -> ( E. n e. { 0 } A = ( P ^ n ) <-> A = ( P ^ 0 ) ) )
13	9, 12	ax-mp 5	. . . . . 6 |- ( E. n e. { 0 } A = ( P ^ n ) <-> A = ( P ^ 0 ) )
14		prmnn 12064	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. NN )
15	14	nncnd 8892	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. CC )
16	15	exp0d 10603	. . . . . . . . . . 11 |- ( P e. Prime -> ( P ^ 0 ) = 1 )
17	16	3ad2ant1 1013	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( P ^ 0 ) = 1 )
18	17	eqeq2d 2182	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A = ( P ^ 0 ) <-> A = 1 ) )
19		eluz2b3 9563	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) <-> ( A e. NN /\ A =/= 1 ) )
20		eqneqall 2350	. . . . . . . . . . . 12 |- ( A = 1 -> ( A =/= 1 -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
21	20	com12 30	. . . . . . . . . . 11 |- ( A =/= 1 -> ( A = 1 -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
22	19, 21	simplbiim 385	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> ( A = 1 -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
23	22	3ad2ant2 1014	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A = 1 -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
24	18, 23	sylbid 149	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A = ( P ^ 0 ) -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
25	24	com12 30	. . . . . . 7 |- ( A = ( P ^ 0 ) -> ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
26	25	impd 252	. . . . . 6 |- ( A = ( P ^ 0 ) -> ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN A = ( P ^ n ) ) )
27	13, 26	sylbi 120	. . . . 5 |- ( E. n e. { 0 } A = ( P ^ n ) -> ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN A = ( P ^ n ) ) )
28	27	jao1i 791	. . . 4 |- ( ( E. n e. NN A = ( P ^ n ) \/ E. n e. { 0 } A = ( P ^ n ) ) -> ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN A = ( P ^ n ) ) )
29	8, 28	sylbi 120	. . 3 |- ( E. n e. NN0 A = ( P ^ n ) -> ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN A = ( P ^ n ) ) )
30	4, 29	mpcom 36	. 2 |- ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN A = ( P ^ n ) )
31	30	ex 114	1 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   E.wrex 2449   u. cun 3119   {csn 3583   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ^cexp 10475   || cdvds 11749   Primecprime 12061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-xnn0 9199  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239

This theorem is referenced by:  difsqpwdvds  12291