Theorem prmlem0 15812

Description: Lemma for prmlem1 15814 and prmlem2 15827. (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
prmlem0.1	|- ( ( -. 2 || M /\ x e. ( ZZ>= ` M ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )
prmlem0.2	|- ( K e. Prime -> -. K || N )
prmlem0.3	|- ( K + 2 ) = M

Assertion

Ref	Expression
prmlem0	|- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )

Distinct variable group:   x, N

Allowed substitution hints:   K( x)   M( x)

Proof of Theorem prmlem0

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . . 5 |- ( x e. ( Prime \ { 2 } ) -> x e. Prime )
2		prmlem0.2	. . . . . 6 |- ( K e. Prime -> -. K || N )
3		eleq1 2689	. . . . . . 7 |- ( x = K -> ( x e. Prime <-> K e. Prime ) )
4		breq1 4656	. . . . . . . 8 |- ( x = K -> ( x || N <-> K || N ) )
5	4	notbid 308	. . . . . . 7 |- ( x = K -> ( -. x || N <-> -. K || N ) )
6	3, 5	imbi12d 334	. . . . . 6 |- ( x = K -> ( ( x e. Prime -> -. x || N ) <-> ( K e. Prime -> -. K || N ) ) )
7	2, 6	mpbiri 248	. . . . 5 |- ( x = K -> ( x e. Prime -> -. x || N ) )
8	1, 7	syl5 34	. . . 4 |- ( x = K -> ( x e. ( Prime \ { 2 } ) -> -. x || N ) )
9	8	adantrd 484	. . 3 |- ( x = K -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )
10	9	a1i 11	. 2 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x = K -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
11		uzp1 11721	. . 3 |- ( x e. ( ZZ>= ` ( K + 1 ) ) -> ( x = ( K + 1 ) \/ x e. ( ZZ>= ` ( ( K + 1 ) + 1 ) ) ) )
12		eleq1 2689	. . . . . . . 8 |- ( x = ( K + 1 ) -> ( x e. ( Prime \ { 2 } ) <-> ( K + 1 ) e. ( Prime \ { 2 } ) ) )
13	12	adantl 482	. . . . . . 7 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( x e. ( Prime \ { 2 } ) <-> ( K + 1 ) e. ( Prime \ { 2 } ) ) )
14		eldifsn 4317	. . . . . . . . 9 |- ( ( K + 1 ) e. ( Prime \ { 2 } ) <-> ( ( K + 1 ) e. Prime /\ ( K + 1 ) =/= 2 ) )
15		eluzel2 11692	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ZZ>= ` K ) -> K e. ZZ )
16	15	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> K e. ZZ )
17		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> -. 2 || K )
18		1z 11407	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
19		n2dvds1 15104	. . . . . . . . . . . . . . . . 17 |- -. 2 || 1
20		opoe 15087	. . . . . . . . . . . . . . . . 17 |- ( ( ( K e. ZZ /\ -. 2 || K ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( K + 1 ) )
21	18, 19, 20	mpanr12 721	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ -. 2 || K ) -> 2 || ( K + 1 ) )
22	16, 17, 21	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> 2 || ( K + 1 ) )
23	22	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> 2 || ( K + 1 ) )
24		2z 11409	. . . . . . . . . . . . . . . 16 |- 2 e. ZZ
25		uzid 11702	. . . . . . . . . . . . . . . 16 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
26	24, 25	mp1i 13	. . . . . . . . . . . . . . 15 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> 2 e. ( ZZ>= ` 2 ) )
27		dvdsprm 15415	. . . . . . . . . . . . . . 15 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ ( K + 1 ) e. Prime ) -> ( 2 || ( K + 1 ) <-> 2 = ( K + 1 ) ) )
28	26, 27	sylan 488	. . . . . . . . . . . . . 14 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( 2 || ( K + 1 ) <-> 2 = ( K + 1 ) ) )
29	23, 28	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> 2 = ( K + 1 ) )
30	29	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( K + 1 ) = 2 )
31	30	a1d 25	. . . . . . . . . . 11 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( x || N -> ( K + 1 ) = 2 ) )
32	31	necon3ad 2807	. . . . . . . . . 10 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( ( K + 1 ) =/= 2 -> -. x || N ) )
33	32	expimpd 629	. . . . . . . . 9 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( ( K + 1 ) e. Prime /\ ( K + 1 ) =/= 2 ) -> -. x || N ) )
34	14, 33	syl5bi 232	. . . . . . . 8 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( K + 1 ) e. ( Prime \ { 2 } ) -> -. x || N ) )
35	34	adantr 481	. . . . . . 7 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( ( K + 1 ) e. ( Prime \ { 2 } ) -> -. x || N ) )
36	13, 35	sylbid 230	. . . . . 6 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( x e. ( Prime \ { 2 } ) -> -. x || N ) )
37	36	adantrd 484	. . . . 5 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )
38	37	ex 450	. . . 4 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x = ( K + 1 ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
39	16	zcnd 11483	. . . . . . . . 9 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> K e. CC )
40		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
41		addass 10023	. . . . . . . . . 10 |- ( ( K e. CC /\ 1 e. CC /\ 1 e. CC ) -> ( ( K + 1 ) + 1 ) = ( K + ( 1 + 1 ) ) )
42	40, 40, 41	mp3an23 1416	. . . . . . . . 9 |- ( K e. CC -> ( ( K + 1 ) + 1 ) = ( K + ( 1 + 1 ) ) )
43	39, 42	syl 17	. . . . . . . 8 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( K + 1 ) + 1 ) = ( K + ( 1 + 1 ) ) )
44		1p1e2 11134	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
45	44	oveq2i 6661	. . . . . . . . 9 |- ( K + ( 1 + 1 ) ) = ( K + 2 )
46		prmlem0.3	. . . . . . . . 9 |- ( K + 2 ) = M
47	45, 46	eqtri 2644	. . . . . . . 8 |- ( K + ( 1 + 1 ) ) = M
48	43, 47	syl6eq 2672	. . . . . . 7 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( K + 1 ) + 1 ) = M )
49	48	fveq2d 6195	. . . . . 6 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ZZ>= ` ( ( K + 1 ) + 1 ) ) = ( ZZ>= ` M ) )
50	49	eleq2d 2687	. . . . 5 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` ( ( K + 1 ) + 1 ) ) <-> x e. ( ZZ>= ` M ) ) )
51		dvdsaddr 15025	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ K e. ZZ ) -> ( 2 || K <-> 2 || ( K + 2 ) ) )
52	24, 16, 51	sylancr 695	. . . . . . . 8 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( 2 || K <-> 2 || ( K + 2 ) ) )
53	46	breq2i 4661	. . . . . . . 8 |- ( 2 || ( K + 2 ) <-> 2 || M )
54	52, 53	syl6bb 276	. . . . . . 7 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( 2 || K <-> 2 || M ) )
55	17, 54	mtbid 314	. . . . . 6 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> -. 2 || M )
56		prmlem0.1	. . . . . . 7 |- ( ( -. 2 || M /\ x e. ( ZZ>= ` M ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )
57	56	ex 450	. . . . . 6 |- ( -. 2 || M -> ( x e. ( ZZ>= ` M ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
58	55, 57	syl 17	. . . . 5 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` M ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
59	50, 58	sylbid 230	. . . 4 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` ( ( K + 1 ) + 1 ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
60	38, 59	jaod 395	. . 3 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( x = ( K + 1 ) \/ x e. ( ZZ>= ` ( ( K + 1 ) + 1 ) ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
61	11, 60	syl5 34	. 2 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` ( K + 1 ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
62		uzp1 11721	. . 3 |- ( x e. ( ZZ>= ` K ) -> ( x = K \/ x e. ( ZZ>= ` ( K + 1 ) ) ) )
63	62	adantl 482	. 2 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x = K \/ x e. ( ZZ>= ` ( K + 1 ) ) ) )
64	10, 61, 63	mpjaod 396	1 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   <_ cle 10075   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   || cdvds 14983   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386

This theorem is referenced by:  prmlem1a  15813  prmlem2  15827