Theorem ruclem1 14960

Description: Lemma for ruc 14972 (the reals are uncountable). Substitutions for the function D. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)

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 ) )

Assertion

Ref	Expression
ruclem1	|- ( 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 ) ) )

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 ruclem1

Step	Hyp	Ref	Expression
1		ruc.2	. . . . . 6 |- ( 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 ) >. ) ) )
2	1	oveqd 6667	. . . . 5 |- ( ph -> ( <. A , B >. D M ) = ( <. A , B >. ( 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 ) >. ) ) M ) )
3		ruclem1.3	. . . . . . 7 |- ( ph -> A e. RR )
4		ruclem1.4	. . . . . . 7 |- ( ph -> B e. RR )
5		opelxpi 5148	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
6	3, 4, 5	syl2anc 693	. . . . . 6 |- ( ph -> <. A , B >. e. ( RR X. RR ) )
7		ruclem1.5	. . . . . 6 |- ( ph -> M e. RR )
8		simpr 477	. . . . . . . . . . 11 |- ( ( x = <. A , B >. /\ y = M ) -> y = M )
9	8	breq2d 4665	. . . . . . . . . 10 |- ( ( x = <. A , B >. /\ y = M ) -> ( m < y <-> m < M ) )
10		simpl 473	. . . . . . . . . . . 12 |- ( ( x = <. A , B >. /\ y = M ) -> x = <. A , B >. )
11	10	fveq2d 6195	. . . . . . . . . . 11 |- ( ( x = <. A , B >. /\ y = M ) -> ( 1st ` x ) = ( 1st ` <. A , B >. ) )
12	11	opeq1d 4408	. . . . . . . . . 10 |- ( ( x = <. A , B >. /\ y = M ) -> <. ( 1st ` x ) , m >. = <. ( 1st ` <. A , B >. ) , m >. )
13	10	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( x = <. A , B >. /\ y = M ) -> ( 2nd ` x ) = ( 2nd ` <. A , B >. ) )
14	13	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( x = <. A , B >. /\ y = M ) -> ( m + ( 2nd ` x ) ) = ( m + ( 2nd ` <. A , B >. ) ) )
15	14	oveq1d 6665	. . . . . . . . . . 11 |- ( ( x = <. A , B >. /\ y = M ) -> ( ( m + ( 2nd ` x ) ) / 2 ) = ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) )
16	15, 13	opeq12d 4410	. . . . . . . . . 10 |- ( ( x = <. A , B >. /\ y = M ) -> <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. = <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. )
17	9, 12, 16	ifbieq12d 4113	. . . . . . . . 9 |- ( ( x = <. A , B >. /\ y = M ) -> if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) = if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
18	17	csbeq2dv 3992	. . . . . . . 8 |- ( ( x = <. A , B >. /\ y = M ) -> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) = [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
19	11, 13	oveq12d 6668	. . . . . . . . . 10 |- ( ( x = <. A , B >. /\ y = M ) -> ( ( 1st ` x ) + ( 2nd ` x ) ) = ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) )
20	19	oveq1d 6665	. . . . . . . . 9 |- ( ( x = <. A , B >. /\ y = M ) -> ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) = ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) )
21	20	csbeq1d 3540	. . . . . . . 8 |- ( ( x = <. A , B >. /\ y = M ) -> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
22	18, 21	eqtrd 2656	. . . . . . 7 |- ( ( x = <. A , B >. /\ y = M ) -> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) = [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
23		eqid 2622	. . . . . . 7 |- ( 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 ) >. ) ) = ( 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 ) >. ) )
24		opex 4932	. . . . . . . . 9 |- <. ( 1st ` <. A , B >. ) , m >. e. _V
25		opex 4932	. . . . . . . . 9 |- <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. e. _V
26	24, 25	ifex 4156	. . . . . . . 8 |- if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) e. _V
27	26	csbex 4793	. . . . . . 7 |- [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) e. _V
28	22, 23, 27	ovmpt2a 6791	. . . . . 6 |- ( ( <. A , B >. e. ( RR X. RR ) /\ M e. RR ) -> ( <. A , B >. ( 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 ) >. ) ) M ) = [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
29	6, 7, 28	syl2anc 693	. . . . 5 |- ( ph -> ( <. A , B >. ( 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 ) >. ) ) M ) = [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
30	2, 29	eqtrd 2656	. . . 4 |- ( ph -> ( <. A , B >. D M ) = [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
31		op1stg 7180	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( 1st ` <. A , B >. ) = A )
32	3, 4, 31	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 1st ` <. A , B >. ) = A )
33		op2ndg 7181	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( 2nd ` <. A , B >. ) = B )
34	3, 4, 33	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 2nd ` <. A , B >. ) = B )
35	32, 34	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) = ( A + B ) )
36	35	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) = ( ( A + B ) / 2 ) )
37	36	csbeq1d 3540	. . . . 5 |- ( ph -> [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = [_ ( ( A + B ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
38		ovex 6678	. . . . . . 7 |- ( ( A + B ) / 2 ) e. _V
39		breq1 4656	. . . . . . . 8 |- ( m = ( ( A + B ) / 2 ) -> ( m < M <-> ( ( A + B ) / 2 ) < M ) )
40		opeq2 4403	. . . . . . . 8 |- ( m = ( ( A + B ) / 2 ) -> <. ( 1st ` <. A , B >. ) , m >. = <. ( 1st ` <. A , B >. ) , ( ( A + B ) / 2 ) >. )
41		oveq1 6657	. . . . . . . . . 10 |- ( m = ( ( A + B ) / 2 ) -> ( m + ( 2nd ` <. A , B >. ) ) = ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) )
42	41	oveq1d 6665	. . . . . . . . 9 |- ( m = ( ( A + B ) / 2 ) -> ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) = ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) )
43	42	opeq1d 4408	. . . . . . . 8 |- ( m = ( ( A + B ) / 2 ) -> <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. = <. ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. )
44	39, 40, 43	ifbieq12d 4113	. . . . . . 7 |- ( m = ( ( A + B ) / 2 ) -> if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = if ( ( ( A + B ) / 2 ) < M , <. ( 1st ` <. A , B >. ) , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) )
45	38, 44	csbie 3559	. . . . . 6 |- [_ ( ( A + B ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = if ( ( ( A + B ) / 2 ) < M , <. ( 1st ` <. A , B >. ) , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. )
46	32	opeq1d 4408	. . . . . . 7 |- ( ph -> <. ( 1st ` <. A , B >. ) , ( ( A + B ) / 2 ) >. = <. A , ( ( A + B ) / 2 ) >. )
47	34	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) = ( ( ( A + B ) / 2 ) + B ) )
48	47	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) = ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
49	48, 34	opeq12d 4410	. . . . . . 7 |- ( ph -> <. ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. = <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. )
50	46, 49	ifeq12d 4106	. . . . . 6 |- ( ph -> if ( ( ( A + B ) / 2 ) < M , <. ( 1st ` <. A , B >. ) , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) )
51	45, 50	syl5eq 2668	. . . . 5 |- ( ph -> [_ ( ( A + B ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) )
52	37, 51	eqtrd 2656	. . . 4 |- ( ph -> [_ ( ( ( 1st ` <. A , B >. ) + ( 2nd ` <. A , B >. ) ) / 2 ) / m ]_ if ( m < M , <. ( 1st ` <. A , B >. ) , m >. , <. ( ( m + ( 2nd ` <. A , B >. ) ) / 2 ) , ( 2nd ` <. A , B >. ) >. ) = if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) )
53	30, 52	eqtrd 2656	. . 3 |- ( ph -> ( <. A , B >. D M ) = if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) )
54	3, 4	readdcld 10069	. . . . . 6 |- ( ph -> ( A + B ) e. RR )
55	54	rehalfcld 11279	. . . . 5 |- ( ph -> ( ( A + B ) / 2 ) e. RR )
56		opelxpi 5148	. . . . 5 |- ( ( A e. RR /\ ( ( A + B ) / 2 ) e. RR ) -> <. A , ( ( A + B ) / 2 ) >. e. ( RR X. RR ) )
57	3, 55, 56	syl2anc 693	. . . 4 |- ( ph -> <. A , ( ( A + B ) / 2 ) >. e. ( RR X. RR ) )
58	55, 4	readdcld 10069	. . . . . 6 |- ( ph -> ( ( ( A + B ) / 2 ) + B ) e. RR )
59	58	rehalfcld 11279	. . . . 5 |- ( ph -> ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. RR )
60		opelxpi 5148	. . . . 5 |- ( ( ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. RR /\ B e. RR ) -> <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. e. ( RR X. RR ) )
61	59, 4, 60	syl2anc 693	. . . 4 |- ( ph -> <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. e. ( RR X. RR ) )
62	57, 61	ifcld 4131	. . 3 |- ( ph -> if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) e. ( RR X. RR ) )
63	53, 62	eqeltrd 2701	. 2 |- ( ph -> ( <. A , B >. D M ) e. ( RR X. RR ) )
64		ruclem1.6	. . 3 |- X = ( 1st ` ( <. A , B >. D M ) )
65	53	fveq2d 6195	. . . 4 |- ( ph -> ( 1st ` ( <. A , B >. D M ) ) = ( 1st ` if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) )
66		fvif 6204	. . . . 5 |- ( 1st ` if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) = if ( ( ( A + B ) / 2 ) < M , ( 1st ` <. A , ( ( A + B ) / 2 ) >. ) , ( 1st ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) )
67		op1stg 7180	. . . . . . 7 |- ( ( A e. RR /\ ( ( A + B ) / 2 ) e. _V ) -> ( 1st ` <. A , ( ( A + B ) / 2 ) >. ) = A )
68	3, 38, 67	sylancl 694	. . . . . 6 |- ( ph -> ( 1st ` <. A , ( ( A + B ) / 2 ) >. ) = A )
69		ovex 6678	. . . . . . 7 |- ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. _V
70		op1stg 7180	. . . . . . 7 |- ( ( ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. _V /\ B e. RR ) -> ( 1st ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) = ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
71	69, 4, 70	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) = ( ( ( ( A + B ) / 2 ) + B ) / 2 ) )
72	68, 71	ifeq12d 4106	. . . . 5 |- ( ph -> if ( ( ( A + B ) / 2 ) < M , ( 1st ` <. A , ( ( A + B ) / 2 ) >. ) , ( 1st ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) = if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
73	66, 72	syl5eq 2668	. . . 4 |- ( ph -> ( 1st ` if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) = if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
74	65, 73	eqtrd 2656	. . 3 |- ( ph -> ( 1st ` ( <. A , B >. D M ) ) = if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
75	64, 74	syl5eq 2668	. 2 |- ( ph -> X = if ( ( ( A + B ) / 2 ) < M , A , ( ( ( ( A + B ) / 2 ) + B ) / 2 ) ) )
76		ruclem1.7	. . 3 |- Y = ( 2nd ` ( <. A , B >. D M ) )
77	53	fveq2d 6195	. . . 4 |- ( ph -> ( 2nd ` ( <. A , B >. D M ) ) = ( 2nd ` if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) )
78		fvif 6204	. . . . 5 |- ( 2nd ` if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) = if ( ( ( A + B ) / 2 ) < M , ( 2nd ` <. A , ( ( A + B ) / 2 ) >. ) , ( 2nd ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) )
79		op2ndg 7181	. . . . . . 7 |- ( ( A e. RR /\ ( ( A + B ) / 2 ) e. _V ) -> ( 2nd ` <. A , ( ( A + B ) / 2 ) >. ) = ( ( A + B ) / 2 ) )
80	3, 38, 79	sylancl 694	. . . . . 6 |- ( ph -> ( 2nd ` <. A , ( ( A + B ) / 2 ) >. ) = ( ( A + B ) / 2 ) )
81		op2ndg 7181	. . . . . . 7 |- ( ( ( ( ( ( A + B ) / 2 ) + B ) / 2 ) e. _V /\ B e. RR ) -> ( 2nd ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) = B )
82	69, 4, 81	sylancr 695	. . . . . 6 |- ( ph -> ( 2nd ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) = B )
83	80, 82	ifeq12d 4106	. . . . 5 |- ( ph -> if ( ( ( A + B ) / 2 ) < M , ( 2nd ` <. A , ( ( A + B ) / 2 ) >. ) , ( 2nd ` <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) = if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) )
84	78, 83	syl5eq 2668	. . . 4 |- ( ph -> ( 2nd ` if ( ( ( A + B ) / 2 ) < M , <. A , ( ( A + B ) / 2 ) >. , <. ( ( ( ( A + B ) / 2 ) + B ) / 2 ) , B >. ) ) = if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) )
85	77, 84	eqtrd 2656	. . 3 |- ( ph -> ( 2nd ` ( <. A , B >. D M ) ) = if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) )
86	76, 85	syl5eq 2668	. 2 |- ( ph -> Y = if ( ( ( A + B ) / 2 ) < M , ( ( A + B ) / 2 ) , B ) )
87	63, 75, 86	3jca 1242	1 |- ( 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_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   / 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:  ruclem2  14961  ruclem3  14962  ruclem6  14964