Theorem pcprendvds 12486

Description: Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Hypotheses

Ref	Expression
pclem.1	|- A = { n e. NN0 | ( P ^ n ) || N }
pclem.2	|- S = sup ( A , RR , < )

Assertion

Ref	Expression
pcprendvds	|- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )

Distinct variable groups:   n, N   P, n

Allowed substitution hints:   A( n)   S( n)

Proof of Theorem pcprendvds

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pclem.1	. . . . . . 7 |- A = { n e. NN0 | ( P ^ n ) || N }
2		pclem.2	. . . . . . 7 |- S = sup ( A , RR , < )
3	1, 2	pcprecl 12485	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
4	3	simpld 112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
5	4	nn0red 9322	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. RR )
6	5	ltp1d 8976	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S < ( S + 1 ) )
7	4	nn0zd 9465	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. ZZ )
8	7	peano2zd 9470	. . . 4 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + 1 ) e. ZZ )
9		zltnle 9391	. . . 4 |- ( ( S e. ZZ /\ ( S + 1 ) e. ZZ ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
10	7, 8, 9	syl2anc 411	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S < ( S + 1 ) <-> -. ( S + 1 ) <_ S ) )
11	6, 10	mpbid 147	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + 1 ) <_ S )
12		peano2nn0 9308	. . . 4 |- ( S e. NN0 -> ( S + 1 ) e. NN0 )
13		oveq2 5933	. . . . . . 7 |- ( x = ( S + 1 ) -> ( P ^ x ) = ( P ^ ( S + 1 ) ) )
14	13	breq1d 4044	. . . . . 6 |- ( x = ( S + 1 ) -> ( ( P ^ x ) || N <-> ( P ^ ( S + 1 ) ) || N ) )
15		oveq2 5933	. . . . . . . . 9 |- ( n = x -> ( P ^ n ) = ( P ^ x ) )
16	15	breq1d 4044	. . . . . . . 8 |- ( n = x -> ( ( P ^ n ) || N <-> ( P ^ x ) || N ) )
17	16	cbvrabv 2762	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || N } = { x e. NN0 | ( P ^ x ) || N }
18	1, 17	eqtri 2217	. . . . . 6 |- A = { x e. NN0 | ( P ^ x ) || N }
19	14, 18	elrab2 2923	. . . . 5 |- ( ( S + 1 ) e. A <-> ( ( S + 1 ) e. NN0 /\ ( P ^ ( S + 1 ) ) || N ) )
20	19	simplbi2 385	. . . 4 |- ( ( S + 1 ) e. NN0 -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) e. A ) )
21	4, 12, 20	3syl 17	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) e. A ) )
22	1	ssrab3 3270	. . . . . . . 8 |- A C_ NN0
23		nn0ssz 9363	. . . . . . . 8 |- NN0 C_ ZZ
24	22, 23	sstri 3193	. . . . . . 7 |- A C_ ZZ
25	24	a1i 9	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> A C_ ZZ )
26	1	pclemdc 12484	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )
27	26	adantr 276	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> A. x e. ZZ DECID x e. A )
28	1	pclemub 12483	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. A y <_ x )
29	28	adantr 276	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> E. x e. ZZ A. y e. A y <_ x )
30		simpr 110	. . . . . 6 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) e. A )
31	25, 27, 29, 30	suprzubdc 10345	. . . . 5 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ sup ( A , RR , < ) )
32	31, 2	breqtrrdi 4076	. . . 4 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( S + 1 ) e. A ) -> ( S + 1 ) <_ S )
33	32	ex 115	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + 1 ) e. A -> ( S + 1 ) <_ S ) )
34	21, 33	syld 45	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + 1 ) ) || N -> ( S + 1 ) <_ S ) )
35	11, 34	mtod 664	1 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( S + 1 ) ) || N )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   E.wrex 2476   {crab 2479   C_ wss 3157   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   supcsup 7057   RRcr 7897   0cc0 7898   1c1 7899   + caddc 7901   < clt 8080   <_ cle 8081   2c2 9060   NN0cn0 9268   ZZcz 9345   ZZ>=cuz 9620   ^cexp 10649   || cdvds 11971

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-inf 7060  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-fz 10103  df-fzo 10237  df-fl 10379  df-mod 10434  df-seqfrec 10559  df-exp 10650  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-dvds 11972

This theorem is referenced by:  pcprendvds2  12487  pczndvds  12512