Theorem pclemdc 12924

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 9889	. . . . . 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 9809	. . . . . . . 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 10862	. . . . . . 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 12436	. . . . . 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 6036	. . . . . . 7 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
13	12	breq1d 4103	. . . . . 6 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
14		pclem.1	. . . . . 6 |- A = { n e. NN0 | ( P ^ n ) || N }
15	13, 14	elrab2 2966	. . . . 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 2606	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 2202   =/= wne 2403   A.wral 2511   {crab 2515   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   0cc0 8075   2c2 9236   NN0cn0 9444   ZZcz 9523   ZZ>=cuz 9799   ^cexp 10846   || cdvds 12411

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-dvds 12412

This theorem is referenced by:  pcprecl  12925  pcprendvds  12926  pcpremul  12929