Theorem pclemdc 12231

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 9559	. . . . . 6 |- ( x e. ZZ -> DECID x e. NN0 )
2	1	ad2antlr 486	. . . . 5 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID x e. NN0 )
3		eluzelz 9485	. . . . . . . 8 |- ( P e. ( ZZ>= ` 2 ) -> P e. ZZ )
4	3	ad3antrrr 489	. . . . . . 7 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> P e. ZZ )
5		simpr 109	. . . . . . 7 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> x e. NN0 )
6		zexpcl 10480	. . . . . . 7 |- ( ( P e. ZZ /\ x e. NN0 ) -> ( P ^ x ) e. ZZ )
7	4, 5, 6	syl2anc 409	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> ( P ^ x ) e. ZZ )
8		simprl 526	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
9	8	ad2antrr 485	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> N e. ZZ )
10		zdvdsdc 11763	. . . . . 6 |- ( ( ( P ^ x ) e. ZZ /\ N e. ZZ ) -> DECID ( P ^ x ) || N )
11	7, 9, 10	syl2anc 409	. . . . 5 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID ( P ^ x ) || N )
12		dcan2 929	. . . . 5 |- (DECID x e. NN0 -> (DECID ( P ^ x ) || N -> DECID ( x e. NN0 /\ ( P ^ x ) || N ) ) )
13	2, 11, 12	sylc 62	. . . 4 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID ( x e. NN0 /\ ( P ^ x ) || N ) )
14		oveq2 5859	. . . . . . 7 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
15	14	breq1d 3997	. . . . . 6 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
16		pclem.1	. . . . . 6 |- A = { n e. NN0 | ( P ^ n ) || N }
17	15, 16	elrab2 2889	. . . . 5 |- ( x e. A <-> ( x e. NN0 /\ ( P ^ x ) || N ) )
18	17	dcbii 835	. . . 4 |- (DECID x e. A <-> DECID ( x e. NN0 /\ ( P ^ x ) || N ) )
19	13, 18	sylibr 133	. . 3 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID x e. A )
20		simpr 109	. . . . . . 7 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> -. x e. NN0 )
21	20	intnanrd 927	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> -. ( x e. NN0 /\ ( P ^ x ) || N ) )
22	21	olcd 729	. . . . 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 ) ) )
23		df-dc 830	. . . . 5 |- (DECID ( x e. NN0 /\ ( P ^ x ) || N ) <-> ( ( x e. NN0 /\ ( P ^ x ) || N ) \/ -. ( x e. NN0 /\ ( P ^ x ) || N ) ) )
24	22, 23	sylibr 133	. . . 4 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> DECID ( x e. NN0 /\ ( P ^ x ) || N ) )
25	24, 18	sylibr 133	. . 3 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ -. x e. NN0 ) -> DECID x e. A )
26		exmiddc 831	. . . . 5 |- (DECID x e. NN0 -> ( x e. NN0 \/ -. x e. NN0 ) )
27	1, 26	syl 14	. . . 4 |- ( x e. ZZ -> ( x e. NN0 \/ -. x e. NN0 ) )
28	27	adantl 275	. . 3 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) -> ( x e. NN0 \/ -. x e. NN0 ) )
29	19, 25, 28	mpjaodan 793	. 2 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) -> DECID x e. A )
30	29	ralrimiva 2543	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 103   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   {crab 2452   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   0cc0 7763   2c2 8918   NN0cn0 9124   ZZcz 9201   ZZ>=cuz 9476   ^cexp 10464   || cdvds 11738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-dvds 11739

This theorem is referenced by:  pcprecl  12232  pcprendvds  12233  pcpremul  12236