Theorem dvdsprmpweqnn 12601

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 9686	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
2		dvdsprmpweq 12600	. . . . 5 |- ( ( P e. Prime /\ A e. NN /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
3	1, 2	syl3an2 1283	. . . 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 9295	. . . . . 6 |- NN0 = ( NN u. { 0 } )
6	5	rexeqi 2706	. . . . 5 |- ( E. n e. NN0 A = ( P ^ n ) <-> E. n e. ( NN u. { 0 } ) A = ( P ^ n ) )
7		rexun 3352	. . . . 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 9382	. . . . . . 7 |- 0 e. ZZ
10		oveq2 5951	. . . . . . . . 9 |- ( n = 0 -> ( P ^ n ) = ( P ^ 0 ) )
11	10	eqeq2d 2216	. . . . . . . 8 |- ( n = 0 -> ( A = ( P ^ n ) <-> A = ( P ^ 0 ) ) )
12	11	rexsng 3673	. . . . . . 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 12374	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. NN )
15	14	nncnd 9049	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. CC )
16	15	exp0d 10810	. . . . . . . . . . 11 |- ( P e. Prime -> ( P ^ 0 ) = 1 )
17	16	3ad2ant1 1020	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( P ^ 0 ) = 1 )
18	17	eqeq2d 2216	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A = ( P ^ 0 ) <-> A = 1 ) )
19		eluz2b3 9724	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) <-> ( A e. NN /\ A =/= 1 ) )
20		eqneqall 2385	. . . . . . . . . . . 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 1021	. . . . . . . . 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 797	. . . 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 709   /\ w3a 980   = wceq 1372   e. wcel 2175   =/= wne 2375   E.wrex 2484   u. cun 3163   {csn 3632   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   0cc0 7924   1c1 7925   NNcn 9035   2c2 9086   NN0cn0 9294   ZZcz 9371   ZZ>=cuz 9647   ^cexp 10681   || cdvds 12040   Primecprime 12371

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-2o 6502  df-er 6619  df-en 6827  df-sup 7085  df-inf 7086  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-xnn0 9358  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-dvds 12041  df-gcd 12217  df-prm 12372  df-pc 12550

This theorem is referenced by:  difsqpwdvds  12603