Theorem pclemdc 12216

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 9545	. . . . . 6 |- ( x e. ZZ -> DECID x e. NN0 )
2	1	ad2antlr 481	. . . . 5 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> DECID x e. NN0 )
3		eluzelz 9471	. . . . . . . 8 |- ( P e. ( ZZ>= ` 2 ) -> P e. ZZ )
4	3	ad3antrrr 484	. . . . . . 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 10466	. . . . . . 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 521	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
9	8	ad2antrr 480	. . . . . 6 |- ( ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ x e. NN0 ) -> N e. ZZ )
10		zdvdsdc 11748	. . . . . 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 924	. . . . 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 5849	. . . . . . 7 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
15	14	breq1d 3991	. . . . . 6 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
16		pclem.1	. . . . . 6 |- A = { n e. NN0 | ( P ^ n ) || N }
17	15, 16	elrab2 2884	. . . . 5 |- ( x e. A <-> ( x e. NN0 /\ ( P ^ x ) || N ) )
18	17	dcbii 830	. . . 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 922	. . . . . 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 724	. . . . 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 825	. . . . 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 826	. . . . 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 788	. 2 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ x e. ZZ ) -> DECID x e. A )
30	29	ralrimiva 2538	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 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2335   A.wral 2443   {crab 2447   class class class wbr 3981   ` cfv 5187  (class class class)co 5841   0cc0 7749   2c2 8904   NN0cn0 9110   ZZcz 9187   ZZ>=cuz 9462   ^cexp 10450   || cdvds 11723

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-dvds 11724

This theorem is referenced by:  pcprecl  12217  pcprendvds  12218  pcpremul  12221