Theorem bpos1lem 16270

Description: Lemma for bpos1 . (Contributed by Mario Carneiro, 12-Mar-2014.)

Hypotheses

Ref	Expression
bpos1.1	|- ( E. p e. Prime ( N < p /\ p <_ ( 2 x. N ) ) -> ph )
bpos1.2	|- ( N e. ( ZZ>= ` P ) -> ph )
bpos1.3	|- P e. Prime
bpos1.4	|- A e. NN0
bpos1.5	|- ( A x. 2 ) = B
bpos1.6	|- A < P
bpos1.7	|- ( P < B \/ P = B )

Assertion

Ref	Expression
bpos1lem	|- ( N e. ( ZZ>= ` A ) -> ph )

Distinct variable groups:   N, p   P, p

Allowed substitution hints:   ph( p)   A( p)   B( p)

Proof of Theorem bpos1lem

Step	Hyp	Ref	Expression
1		bpos1.3	. . . . . 6 |- P e. Prime
2		prmnn 12907	. . . . . 6 |- ( P e. Prime -> P e. NN )
3	1, 2	ax-mp 5	. . . . 5 |- P e. NN
4	3	nnzi 9670	. . . 4 |- P e. ZZ
5		eluzelz 9941	. . . 4 |- ( N e. ( ZZ>= ` A ) -> N e. ZZ )
6		eluz 9945	. . . 4 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` P ) <-> P <_ N ) )
7	4, 5, 6	sylancr 418	. . 3 |- ( N e. ( ZZ>= ` A ) -> ( N e. ( ZZ>= ` P ) <-> P <_ N ) )
8		bpos1.2	. . 3 |- ( N e. ( ZZ>= ` P ) -> ph )
9	7, 8	biimtrrdi 164	. 2 |- ( N e. ( ZZ>= ` A ) -> ( P <_ N -> ph ) )
10	3	nnrei 9316	. . . . . . . 8 |- P e. RR
11	10	a1i 9	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> P e. RR )
12		bpos1.5	. . . . . . . . 9 |- ( A x. 2 ) = B
13		bpos1.4	. . . . . . . . . . 11 |- A e. NN0
14	13	nn0rei 9579	. . . . . . . . . 10 |- A e. RR
15		2re 9377	. . . . . . . . . 10 |- 2 e. RR
16	14, 15	remulcli 8341	. . . . . . . . 9 |- ( A x. 2 ) e. RR
17	12, 16	eqeltrri 2312	. . . . . . . 8 |- B e. RR
18	17	a1i 9	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> B e. RR )
19		eluzelre 9942	. . . . . . . 8 |- ( N e. ( ZZ>= ` A ) -> N e. RR )
20		remulcl 8308	. . . . . . . 8 |- ( ( 2 e. RR /\ N e. RR ) -> ( 2 x. N ) e. RR )
21	15, 19, 20	sylancr 418	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> ( 2 x. N ) e. RR )
22		bpos1.7	. . . . . . . . 9 |- ( P < B \/ P = B )
23		2nn0 9585	. . . . . . . . . . . . 13 |- 2 e. NN0
24	13, 23	nn0mulcli 9606	. . . . . . . . . . . 12 |- ( A x. 2 ) e. NN0
25	12, 24	eqeltrri 2312	. . . . . . . . . . 11 |- B e. NN0
26	25	nn0zi 9671	. . . . . . . . . 10 |- B e. ZZ
27		zleloe 9696	. . . . . . . . . 10 |- ( ( P e. ZZ /\ B e. ZZ ) -> ( P <_ B <-> ( P < B \/ P = B ) ) )
28	4, 26, 27	mp2an 430	. . . . . . . . 9 |- ( P <_ B <-> ( P < B \/ P = B ) )
29	22, 28	mpbir 146	. . . . . . . 8 |- P <_ B
30	29	a1i 9	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> P <_ B )
31	13	nn0cni 9580	. . . . . . . . 9 |- A e. CC
32		2cn 9378	. . . . . . . . 9 |- 2 e. CC
33	31, 32, 12	mulcomli 8334	. . . . . . . 8 |- ( 2 x. A ) = B
34		eluzle 9944	. . . . . . . . 9 |- ( N e. ( ZZ>= ` A ) -> A <_ N )
35		2pos 9398	. . . . . . . . . . . 12 |- 0 < 2
36	15, 35	pm3.2i 272	. . . . . . . . . . 11 |- ( 2 e. RR /\ 0 < 2 )
37		lemul2 9190	. . . . . . . . . . 11 |- ( ( A e. RR /\ N e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A <_ N <-> ( 2 x. A ) <_ ( 2 x. N ) ) )
38	14, 36, 37	mp3an13 1369	. . . . . . . . . 10 |- ( N e. RR -> ( A <_ N <-> ( 2 x. A ) <_ ( 2 x. N ) ) )
39	19, 38	syl 14	. . . . . . . . 9 |- ( N e. ( ZZ>= ` A ) -> ( A <_ N <-> ( 2 x. A ) <_ ( 2 x. N ) ) )
40	34, 39	mpbid 147	. . . . . . . 8 |- ( N e. ( ZZ>= ` A ) -> ( 2 x. A ) <_ ( 2 x. N ) )
41	33, 40	eqbrtrrid 4166	. . . . . . 7 |- ( N e. ( ZZ>= ` A ) -> B <_ ( 2 x. N ) )
42	11, 18, 21, 30, 41	letrd 8452	. . . . . 6 |- ( N e. ( ZZ>= ` A ) -> P <_ ( 2 x. N ) )
43	42	anim2i 342	. . . . 5 |- ( ( N < P /\ N e. ( ZZ>= ` A ) ) -> ( N < P /\ P <_ ( 2 x. N ) ) )
44		breq2 4134	. . . . . . 7 |- ( p = P -> ( N < p <-> N < P ) )
45		breq1 4133	. . . . . . 7 |- ( p = P -> ( p <_ ( 2 x. N ) <-> P <_ ( 2 x. N ) ) )
46	44, 45	anbi12d 477	. . . . . 6 |- ( p = P -> ( ( N < p /\ p <_ ( 2 x. N ) ) <-> ( N < P /\ P <_ ( 2 x. N ) ) ) )
47	46	rspcev 2929	. . . . 5 |- ( ( P e. Prime /\ ( N < P /\ P <_ ( 2 x. N ) ) ) -> E. p e. Prime ( N < p /\ p <_ ( 2 x. N ) ) )
48	1, 43, 47	sylancr 418	. . . 4 |- ( ( N < P /\ N e. ( ZZ>= ` A ) ) -> E. p e. Prime ( N < p /\ p <_ ( 2 x. N ) ) )
49		bpos1.1	. . . 4 |- ( E. p e. Prime ( N < p /\ p <_ ( 2 x. N ) ) -> ph )
50	48, 49	syl 14	. . 3 |- ( ( N < P /\ N e. ( ZZ>= ` A ) ) -> ph )
51	50	expcom 116	. 2 |- ( N e. ( ZZ>= ` A ) -> ( N < P -> ph ) )
52		zlelttric 9694	. . 3 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( P <_ N \/ N < P ) )
53	4, 5, 52	sylancr 418	. 2 |- ( N e. ( ZZ>= ` A ) -> ( P <_ N \/ N < P ) )
54	9, 51, 53	mpjaod 730	1 |- ( N e. ( ZZ>= ` A ) -> ph )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8179   0cc0 8180   x. cmul 8185   < clt 8361   <_ cle 8362   NNcn 9307   2c2 9358   NN0cn0 9568   ZZcz 9649   ZZ>=cuz 9931   Primecprime 12904

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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296  ax-pre-mulgt0 8297

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  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-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  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-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-2 9366  df-n0 9569  df-z 9650  df-uz 9932  df-prm 12905

This theorem is used by:  bpos1  16271