Theorem dvdsprmpweqnn 12908

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 9799	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
2		dvdsprmpweq 12907	. . . . 5 |- ( ( P e. Prime /\ A e. NN /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
3	1, 2	syl3an2 1307	. . . 4 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
4	3	imp 124	. . 3 |- ( ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) /\ A || ( P ^ N ) ) -> E. n e. NN0 A = ( P ^ n ) )
5		df-n0 9402	. . . . . 6 |- NN0 = ( NN u. { 0 } )
6	5	rexeqi 2735	. . . . 5 |- ( E. n e. NN0 A = ( P ^ n ) <-> E. n e. ( NN u. { 0 } ) A = ( P ^ n ) )
7		rexun 3387	. . . . 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 184	. . . 4 |- ( E. n e. NN0 A = ( P ^ n ) <-> ( E. n e. NN A = ( P ^ n ) \/ E. n e. { 0 } A = ( P ^ n ) ) )
9		0z 9489	. . . . . . 7 |- 0 e. ZZ
10		oveq2 6025	. . . . . . . . 9 |- ( n = 0 -> ( P ^ n ) = ( P ^ 0 ) )
11	10	eqeq2d 2243	. . . . . . . 8 |- ( n = 0 -> ( A = ( P ^ n ) <-> A = ( P ^ 0 ) ) )
12	11	rexsng 3710	. . . . . . 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 12681	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. NN )
15	14	nncnd 9156	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. CC )
16	15	exp0d 10928	. . . . . . . . . . 11 |- ( P e. Prime -> ( P ^ 0 ) = 1 )
17	16	3ad2ant1 1044	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( P ^ 0 ) = 1 )
18	17	eqeq2d 2243	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A = ( P ^ 0 ) <-> A = 1 ) )
19		eluz2b3 9837	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) <-> ( A e. NN /\ A =/= 1 ) )
20		eqneqall 2412	. . . . . . . . . . . 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 387	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> ( A = 1 -> ( A || ( P ^ N ) -> E. n e. NN A = ( P ^ n ) ) ) )
23	22	3ad2ant2 1045	. . . . . . . . 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 150	. . . . . . . 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 254	. . . . . 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 121	. . . . 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 803	. . . 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 121	. . 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 115	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 104   <-> wb 105   \/ wo 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   u. cun 3198   {csn 3669   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   NNcn 9142   2c2 9193   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   ^cexp 10799   || cdvds 12347   Primecprime 12678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-xnn0 9465  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-prm 12679  df-pc 12857

This theorem is referenced by:  difsqpwdvds  12910