Theorem pclemdc 13011

Description: Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)

Hypothesis

Ref	Expression
pclem.1	|- A = { n e. NN0 | ( P ^ n ) || N }

Assertion

Ref	Expression
pclemdc	|- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )

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

Allowed substitution hints:   A( x, n)

Proof of Theorem pclemdc

Step	Hyp	Ref	Expression
1		elnn0dc 9961	. . . . . 6 |- ( x e. ZZ -> DECID x e. NN0 )
2	1	ad2antlr 489	. . . . 5 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID x e. NN0 )
3		eluzelz 9881	. . . . . . . 8 |- ( P e. ( ZZ>= ` 2 ) -> P e. ZZ )
4	3	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> P e. ZZ )
5		zexpcl 10940	. . . . . . 7 |- ( ( P e. ZZ /\ x e. NN0 ) -> ( P ^ x ) e. ZZ )
6	4, 5	sylancom 420	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> ( P ^ x ) e. ZZ )
7		simprl 531	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
8	7	ad2antrr 488	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> N e. ZZ )
9		zdvdsdc 12523	. . . . . 6 |- ( ( ( P ^ x ) e. ZZ /\ N e. ZZ ) -> DECID ( P ^ x ) || N )
10	6, 8, 9	syl2anc 411	. . . . 5 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID ( P ^ x ) || N )
11	2, 10	dcand 941	. . . 4 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID ( x e. NN0 /\ ( P ^ x ) || N ) )
12		oveq2 6066	. . . . . . 7 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
13	12	breq1d 4124	. . . . . 6 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
14		pclem.1	. . . . . 6 |- A = { n e. NN0 | ( P ^ n ) || N }
15	13, 14	elrab2 2979	. . . . 5 |- ( x e. A <-> ( x e. NN0 /\ ( P ^ x ) || N ) )
16	15	dcbii 848	. . . 4 |- (DECID x e. A <-> DECID ( x e. NN0 /\ ( P ^ x ) || N ) )
17	11, 16	sylibr 134	. . 3 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID x e. A )
18		simpr 110	. . . . . . 7 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> -. x e. NN0 )
19	18	intnanrd 940	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> -. ( x e. NN0 /\ ( P ^ x ) || N ) )
20	19	olcd 742	. . . . 5 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> ( ( x e. NN0 /\ ( P ^ x ) || N ) \/ -. ( x e. NN0 /\ ( P ^ x ) || N ) ) )
21		df-dc 843	. . . . 5 |- (DECID ( x e. NN0 /\ ( P ^ x ) || N ) <-> ( ( x e. NN0 /\ ( P ^ x ) || N ) \/ -. ( x e. NN0 /\ ( P ^ x ) || N ) ) )
22	20, 21	sylibr 134	. . . 4 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> DECID ( x e. NN0 /\ ( P ^ x ) || N ) )
23	22, 16	sylibr 134	. . 3 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> DECID x e. A )
24		exmiddc 844	. . . . 5 |- (DECID x e. NN0 -> ( x e. NN0 \/ -. x e. NN0 ) )
25	1, 24	syl 14	. . . 4 |- ( x e. ZZ -> ( x e. NN0 \/ -. x e. NN0 ) )
26	25	adantl 277	. . 3 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) -> ( x e. NN0 \/ -. x e. NN0 ) )
27	17, 23, 26	mpjaodan 806	. 2 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) -> DECID x e. A )
28	27	ralrimiva 2617	1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   {crab 2526   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   2c2 9305   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ^cexp 10924   || cdvds 12498

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-dvds 12499

This theorem is referenced by:  pcprecl  13012  pcprendvds  13013  pcpremul  13016