Theorem prmlem0 13240

Description: Lemma for prmlem1a 13241 and prmlem2 13254. (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 3351	. . . . 5 |- ( x e. ( Prime \ { 2 } ) -> x e. Prime )
2		prmlem0.2	. . . . . 6 |- ( K e. Prime -> -. K || N )
3		eleq1 2301	. . . . . . 7 |- ( x = K -> ( x e. Prime <-> K e. Prime ) )
4		breq1 4133	. . . . . . . 8 |- ( x = K -> ( x || N <-> K || N ) )
5	4	notbid 677	. . . . . . 7 |- ( x = K -> ( -. x || N <-> -. K || N ) )
6	3, 5	imbi12d 234	. . . . . 6 |- ( x = K -> ( ( x e. Prime -> -. x || N ) <-> ( K e. Prime -> -. K || N ) ) )
7	2, 6	mpbiri 168	. . . . 5 |- ( x = K -> ( x e. Prime -> -. x || N ) )
8	1, 7	syl5 32	. . . 4 |- ( x = K -> ( x e. ( Prime \ { 2 } ) -> -. x || N ) )
9	8	adantrd 279	. . 3 |- ( x = K -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )
10	9	a1i 9	. 2 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x = K -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
11		uzp1 9965	. . 3 |- ( x e. ( ZZ>= ` ( K + 1 ) ) -> ( x = ( K + 1 ) \/ x e. ( ZZ>= ` ( ( K + 1 ) + 1 ) ) ) )
12		eleq1 2301	. . . . . . . 8 |- ( x = ( K + 1 ) -> ( x e. ( Prime \ { 2 } ) <-> ( K + 1 ) e. ( Prime \ { 2 } ) ) )
13	12	adantl 277	. . . . . . 7 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( x e. ( Prime \ { 2 } ) <-> ( K + 1 ) e. ( Prime \ { 2 } ) ) )
14		eldifsn 3841	. . . . . . . . 9 |- ( ( K + 1 ) e. ( Prime \ { 2 } ) <-> ( ( K + 1 ) e. Prime /\ ( K + 1 ) =/= 2 ) )
15		eluzel2 9935	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ZZ>= ` K ) -> K e. ZZ )
16	15	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> K e. ZZ )
17		simpl 109	. . . . . . . . . . . . . . . 16 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> -. 2 || K )
18		1z 9674	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
19		n2dvds1 12695	. . . . . . . . . . . . . . . . 17 |- -. 2 || 1
20		opoe 12678	. . . . . . . . . . . . . . . . 17 |- ( ( ( K e. ZZ /\ -. 2 || K ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( K + 1 ) )
21	18, 19, 20	mpanr12 443	. . . . . . . . . . . . . . . 16 |- ( ( K e. ZZ /\ -. 2 || K ) -> 2 || ( K + 1 ) )
22	16, 17, 21	syl2anc 415	. . . . . . . . . . . . . . 15 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> 2 || ( K + 1 ) )
23	22	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> 2 || ( K + 1 ) )
24		2z 9676	. . . . . . . . . . . . . . . 16 |- 2 e. ZZ
25		uzid 9945	. . . . . . . . . . . . . . . 16 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
26	24, 25	mp1i 10	. . . . . . . . . . . . . . 15 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> 2 e. ( ZZ>= ` 2 ) )
27		dvdsprm 12932	. . . . . . . . . . . . . . 15 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ ( K + 1 ) e. Prime ) -> ( 2 || ( K + 1 ) <-> 2 = ( K + 1 ) ) )
28	26, 27	sylan 283	. . . . . . . . . . . . . 14 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( 2 || ( K + 1 ) <-> 2 = ( K + 1 ) ) )
29	23, 28	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> 2 = ( K + 1 ) )
30	29	eqcomd 2244	. . . . . . . . . . . 12 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( K + 1 ) = 2 )
31	30	a1d 22	. . . . . . . . . . 11 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( x || N -> ( K + 1 ) = 2 ) )
32	31	necon3ad 2462	. . . . . . . . . 10 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ ( K + 1 ) e. Prime ) -> ( ( K + 1 ) =/= 2 -> -. x || N ) )
33	32	expimpd 363	. . . . . . . . 9 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( ( K + 1 ) e. Prime /\ ( K + 1 ) =/= 2 ) -> -. x || N ) )
34	14, 33	biimtrid 152	. . . . . . . 8 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( K + 1 ) e. ( Prime \ { 2 } ) -> -. x || N ) )
35	34	adantr 276	. . . . . . 7 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( ( K + 1 ) e. ( Prime \ { 2 } ) -> -. x || N ) )
36	13, 35	sylbid 150	. . . . . 6 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( x e. ( Prime \ { 2 } ) -> -. x || N ) )
37	36	adantrd 279	. . . . 5 |- ( ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) /\ x = ( K + 1 ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )
38	37	ex 115	. . . 4 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x = ( K + 1 ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
39	16	zcnd 9773	. . . . . . . . 9 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> K e. CC )
40		ax-1cn 8272	. . . . . . . . . 10 |- 1 e. CC
41		addass 8309	. . . . . . . . . 10 |- ( ( K e. CC /\ 1 e. CC /\ 1 e. CC ) -> ( ( K + 1 ) + 1 ) = ( K + ( 1 + 1 ) ) )
42	40, 40, 41	mp3an23 1370	. . . . . . . . 9 |- ( K e. CC -> ( ( K + 1 ) + 1 ) = ( K + ( 1 + 1 ) ) )
43	39, 42	syl 14	. . . . . . . 8 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( K + 1 ) + 1 ) = ( K + ( 1 + 1 ) ) )
44		1p1e2 9423	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
45	44	oveq2i 6096	. . . . . . . . 9 |- ( K + ( 1 + 1 ) ) = ( K + 2 )
46		prmlem0.3	. . . . . . . . 9 |- ( K + 2 ) = M
47	45, 46	eqtri 2259	. . . . . . . 8 |- ( K + ( 1 + 1 ) ) = M
48	43, 47	eqtrdi 2287	. . . . . . 7 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( K + 1 ) + 1 ) = M )
49	48	fveq2d 5699	. . . . . 6 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ZZ>= ` ( ( K + 1 ) + 1 ) ) = ( ZZ>= ` M ) )
50	49	eleq2d 2308	. . . . 5 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` ( ( K + 1 ) + 1 ) ) <-> x e. ( ZZ>= ` M ) ) )
51		dvdsaddr 12620	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ K e. ZZ ) -> ( 2 || K <-> 2 || ( K + 2 ) ) )
52	24, 16, 51	sylancr 418	. . . . . . . 8 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( 2 || K <-> 2 || ( K + 2 ) ) )
53	46	breq2i 4138	. . . . . . . 8 |- ( 2 || ( K + 2 ) <-> 2 || M )
54	52, 53	bitrdi 196	. . . . . . 7 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( 2 || K <-> 2 || M ) )
55	17, 54	mtbid 683	. . . . . 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 115	. . . . . 6 |- ( -. 2 || M -> ( x e. ( ZZ>= ` M ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
58	55, 57	syl 14	. . . . 5 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` M ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
59	50, 58	sylbid 150	. . . 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 729	. . 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 32	. 2 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x e. ( ZZ>= ` ( K + 1 ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) ) )
62		uzp1 9965	. . 3 |- ( x e. ( ZZ>= ` K ) -> ( x = K \/ x e. ( ZZ>= ` ( K + 1 ) ) ) )
63	62	adantl 277	. 2 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( x = K \/ x e. ( ZZ>= ` ( K + 1 ) ) ) )
64	10, 61, 63	mpjaod 730	1 |- ( ( -. 2 || K /\ x e. ( ZZ>= ` K ) ) -> ( ( x e. ( Prime \ { 2 } ) /\ ( x ^ 2 ) <_ N ) -> -. x || N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   \ cdif 3217   {csn 3709   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   1c1 8180   + caddc 8182   <_ cle 8361   2c2 9357   ZZcz 9648   ZZ>=cuz 9930   ^cexp 10988   || cdvds 12570   Primecprime 12901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902

This theorem is used by:  prmlem1a  13241  prmlem2  13254