Theorem dvdsprmpweqnn 12734

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 9707	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
2		dvdsprmpweq 12733	. . . . 5 |- ( ( P e. Prime /\ A e. NN /\ N e. NN0 ) -> ( A || ( P ^ N ) -> E. n e. NN0 A = ( P ^ n ) ) )
3	1, 2	syl3an2 1284	. . . 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 9316	. . . . . 6 |- NN0 = ( NN u. { 0 } )
6	5	rexeqi 2708	. . . . 5 |- ( E. n e. NN0 A = ( P ^ n ) <-> E. n e. ( NN u. { 0 } ) A = ( P ^ n ) )
7		rexun 3357	. . . . 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 9403	. . . . . . 7 |- 0 e. ZZ
10		oveq2 5965	. . . . . . . . 9 |- ( n = 0 -> ( P ^ n ) = ( P ^ 0 ) )
11	10	eqeq2d 2218	. . . . . . . 8 |- ( n = 0 -> ( A = ( P ^ n ) <-> A = ( P ^ 0 ) ) )
12	11	rexsng 3679	. . . . . . 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 12507	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. NN )
15	14	nncnd 9070	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. CC )
16	15	exp0d 10834	. . . . . . . . . . 11 |- ( P e. Prime -> ( P ^ 0 ) = 1 )
17	16	3ad2ant1 1021	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( P ^ 0 ) = 1 )
18	17	eqeq2d 2218	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ( ZZ>= ` 2 ) /\ N e. NN0 ) -> ( A = ( P ^ 0 ) <-> A = 1 ) )
19		eluz2b3 9745	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) <-> ( A e. NN /\ A =/= 1 ) )
20		eqneqall 2387	. . . . . . . . . . . 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 1022	. . . . . . . . 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 798	. . . 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 710   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   E.wrex 2486   u. cun 3168   {csn 3638   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   0cc0 7945   1c1 7946   NNcn 9056   2c2 9107   NN0cn0 9315   ZZcz 9392   ZZ>=cuz 9668   ^cexp 10705   || cdvds 12173   Primecprime 12504

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-1o 6515  df-2o 6516  df-er 6633  df-en 6841  df-sup 7101  df-inf 7102  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-xnn0 9379  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-dvds 12174  df-gcd 12350  df-prm 12505  df-pc 12683

This theorem is referenced by:  difsqpwdvds  12736