Theorem pclemdc 12851

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 9835	. . . . . 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 9755	. . . . . . . 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 10806	. . . . . . 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 529	. . . . . . 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 12363	. . . . . 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 938	. . . 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 6021	. . . . . . 7 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
13	12	breq1d 4096	. . . . . 6 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
14		pclem.1	. . . . . 6 |- A = { n e. NN0 | ( P ^ n ) || N }
15	13, 14	elrab2 2963	. . . . 5 |- ( x e. A <-> ( x e. NN0 /\ ( P ^ x ) || N ) )
16	15	dcbii 845	. . . 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 937	. . . . . 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 739	. . . . 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 840	. . . . 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 841	. . . . 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 803	. 2 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) -> DECID x e. A )
28	27	ralrimiva 2603	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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   {crab 2512   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   0cc0 8022   2c2 9184   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   ^cexp 10790   || cdvds 12338

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-dvds 12339

This theorem is referenced by:  pcprecl  12852  pcprendvds  12853  pcpremul  12856