Theorem ruclem3 14962

Description: Lemma for ruc 14972. The constructed interval [ X , Y ] always excludes M. (Contributed by Mario Carneiro, 28-May-2014.)

Hypotheses

Ref	Expression
ruc.1	|- ( ph -> F : NN --> RR )
ruc.2	|- ( ph -> D = ( x e. ( RR X. RR ) , y e. RR |-> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) ) )
ruclem1.3	|- ( ph -> A e. RR )
ruclem1.4	|- ( ph -> B e. RR )
ruclem1.5	|- ( ph -> M e. RR )
ruclem1.6	|- X = ( 1st ` ( <. A , B >. D M ) )
ruclem1.7	|- Y = ( 2nd ` ( <. A , B >. D M ) )
ruclem2.8	|- ( ph -> A < B )

Assertion

Ref	Expression
ruclem3	|- ( ph -> ( M < X \/ Y < M ) )

Distinct variable groups:   x, m, y, A   B, m, x, y   m, F, x, y   m, M, x, y

Allowed substitution hints:   ph( x, y, m)   D( x, y, m)   X( x, y, m)   Y( x, y, m)

Proof of Theorem ruclem3

Step	Hyp	Ref	Expression
1		ruclem1.5	. . . . . . . . 9 |- ( ph -> M e. RR )
2		ruclem1.3	. . . . . . . . . . 11 |- ( ph -> A e. RR )
3		ruclem1.4	. . . . . . . . . . 11 |- ( ph -> B e. RR )
4	2, 3	readdcld 10069	. . . . . . . . . 10 |- ( ph -> ( A + B ) e. RR )
5	4	rehalfcld 11279	. . . . . . . . 9 |- ( ph -> ( ( A + B ) / 2 ) e. RR )
6	1, 5	lenltd 10183	. . . . . . . 8 |- ( ph -> ( M <_ ( ( A + B ) / 2 ) <-> -. ( ( A + B ) / 2 ) < M ) )
7		ruclem2.8	. . . . . . . . . . 11 |- ( ph -> A < B )
8		avglt2 11271	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( ( A + B ) / 2 ) < B ) )
9	2, 3, 8	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( A < B <-> ( ( A + B ) / 2 ) < B ) )
10	7, 9	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( ( A + B ) / 2 ) < B )
11		avglt1 11270	. . . . . . . . . . 11 |- ( ( ( ( A + B ) / 2 ) e. RR /\ B e. RR ) -> ( ( ( A + B ) / 2 ) < B <-> ( ( A + B ) / 2 ) < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
12	5, 3, 11	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( ( A + B ) / 2 ) < B <-> ( ( A + B ) / 2 ) < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
13	10, 12	mpbid 222	. . . . . . . . 9 |- ( ph -> ( ( A + B ) / 2 ) < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
14	5, 3	readdcld 10069	. . . . . . . . . . 11 |- ( ph -> ( ( ( A + B ) / 2 ) + B ) e. RR )
15	14	rehalfcld 11279	. . . . . . . . . 10 |- ( ph -> ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. RR )
16		lelttr 10128	. . . . . . . . . 10 |- ( ( M e. RR /\ ( ( A + B ) / 2 ) e. RR /\ ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. RR ) -> ( ( M <_ ( ( A + B ) / 2 ) /\ ( ( A + B ) / 2 ) < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) -> M < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
17	1, 5, 15, 16	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( M <_ ( ( A + B ) / 2 ) /\ ( ( A + B ) / 2 ) < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) -> M < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
18	13, 17	mpan2d 710	. . . . . . . 8 |- ( ph -> ( M <_ ( ( A + B ) / 2 ) -> M < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
19	6, 18	sylbird 250	. . . . . . 7 |- ( ph -> ( -. ( ( A + B ) / 2 ) < M -> M < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
20	19	imp 445	. . . . . 6 |- ( ( ph /\ -. ( ( A + B ) / 2 ) < M ) -> M < ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
21		ruc.1	. . . . . . . . 9 |- ( ph -> F : NN --> RR )
22		ruc.2	. . . . . . . . 9 |- ( ph -> D = ( x e. ( RR X. RR ) , y e. RR |-> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) ) )
23		ruclem1.6	. . . . . . . . 9 |- X = ( 1st ` ( <. A , B >. D M ) )
24		ruclem1.7	. . . . . . . . 9 |- Y = ( 2nd ` ( <. A , B >. D M ) )
25	21, 22, 2, 3, 1, 23, 24	ruclem1 14960	. . . . . . . 8 |- ( ph -> ( ( <. A , B >. D M ) e. ( RR X. RR ) /\ X = if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) /\ Y = if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) ) )
26	25	simp2d 1074	. . . . . . 7 |- ( ph -> X = if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
27		iffalse 4095	. . . . . . 7 |- ( -. ( ( A + B ) / 2 ) < M -> if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) = ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
28	26, 27	sylan9eq 2676	. . . . . 6 |- ( ( ph /\ -. ( ( A + B ) / 2 ) < M ) -> X = ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
29	20, 28	breqtrrd 4681	. . . . 5 |- ( ( ph /\ -. ( ( A + B ) / 2 ) < M ) -> M < X )
30	29	ex 450	. . . 4 |- ( ph -> ( -. ( ( A + B ) / 2 ) < M -> M < X ) )
31	30	con1d 139	. . 3 |- ( ph -> ( -. M < X -> ( ( A + B ) / 2 ) < M ) )
32	25	simp3d 1075	. . . . . 6 |- ( ph -> Y = if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) )
33		iftrue 4092	. . . . . 6 |- ( ( ( A + B ) / 2 ) < M -> if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) = ( ( A + B ) / 2 ) )
34	32, 33	sylan9eq 2676	. . . . 5 |- ( ( ph /\ ( ( A + B ) / 2 ) < M ) -> Y = ( ( A + B ) / 2 ) )
35		simpr 477	. . . . 5 |- ( ( ph /\ ( ( A + B ) / 2 ) < M ) -> ( ( A + B ) / 2 ) < M )
36	34, 35	eqbrtrd 4675	. . . 4 |- ( ( ph /\ ( ( A + B ) / 2 ) < M ) -> Y < M )
37	36	ex 450	. . 3 |- ( ph -> ( ( ( A + B ) / 2 ) < M -> Y < M ) )
38	31, 37	syld 47	. 2 |- ( ph -> ( -. M < X -> Y < M ) )
39	38	orrd 393	1 |- ( ph -> ( M < X \/ Y < M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   [_csb 3533   ifcif 4086   <.cop 4183   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   + caddc 9939   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-2 11079

This theorem is referenced by:  ruclem12  14970