Theorem ruclem6 14964

Description: Lemma for ruc 14972. Domain and range of the interval sequence. (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 ) >. ) ) )
ruc.4	|- C = ( { <. 0 , <. 0 , 1 >. >. } u. F )
ruc.5	|- G = seq 0 ( D , C )

Assertion

Ref	Expression
ruclem6	|- ( ph -> G : NN0 --> ( RR X. RR ) )

Distinct variable groups:   x, m, y, F   m, G, x, y

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

Proof of Theorem ruclem6

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ruc.5	. . . . . . 7 |- G = seq 0 ( D , C )
2	1	fveq1i 6192	. . . . . 6 |- ( G ` 0 ) = ( seq 0 ( D , C ) ` 0 )
3		0z 11388	. . . . . . 7 |- 0 e. ZZ
4		seq1 12814	. . . . . . 7 |- ( 0 e. ZZ -> ( seq 0 ( D , C ) ` 0 ) = ( C ` 0 ) )
5	3, 4	ax-mp 5	. . . . . 6 |- ( seq 0 ( D , C ) ` 0 ) = ( C ` 0 )
6	2, 5	eqtri 2644	. . . . 5 |- ( G ` 0 ) = ( C ` 0 )
7		ruc.1	. . . . . 6 |- ( ph -> F : NN --> RR )
8		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 ) >. ) ) )
9		ruc.4	. . . . . 6 |- C = ( { <. 0 , <. 0 , 1 >. >. } u. F )
10	7, 8, 9, 1	ruclem4 14963	. . . . 5 |- ( ph -> ( G ` 0 ) = <. 0 , 1 >. )
11	6, 10	syl5eqr 2670	. . . 4 |- ( ph -> ( C ` 0 ) = <. 0 , 1 >. )
12		0re 10040	. . . . 5 |- 0 e. RR
13		1re 10039	. . . . 5 |- 1 e. RR
14		opelxpi 5148	. . . . 5 |- ( ( 0 e. RR /\ 1 e. RR ) -> <. 0 , 1 >. e. ( RR X. RR ) )
15	12, 13, 14	mp2an 708	. . . 4 |- <. 0 , 1 >. e. ( RR X. RR )
16	11, 15	syl6eqel 2709	. . 3 |- ( ph -> ( C ` 0 ) e. ( RR X. RR ) )
17		1st2nd2 7205	. . . . . 6 |- ( z e. ( RR X. RR ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
18	17	ad2antrl 764	. . . . 5 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
19	18	oveq1d 6665	. . . 4 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> ( z D w ) = ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) )
20	7	adantr 481	. . . . . 6 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> F : NN --> RR )
21	8	adantr 481	. . . . . 6 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> 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 ) >. ) ) )
22		xp1st 7198	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. RR )
23	22	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> ( 1st ` z ) e. RR )
24		xp2nd 7199	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. RR )
25	24	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> ( 2nd ` z ) e. RR )
26		simprr 796	. . . . . 6 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> w e. RR )
27		eqid 2622	. . . . . 6 |- ( 1st ` ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) ) = ( 1st ` ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) )
28		eqid 2622	. . . . . 6 |- ( 2nd ` ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) ) = ( 2nd ` ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) )
29	20, 21, 23, 25, 26, 27, 28	ruclem1 14960	. . . . 5 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) e. ( RR X. RR ) /\ ( 1st ` ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) ) = if ( ( ( ( 1st ` z ) + ( 2nd ` z ) ) / 2 ) < w , ( 1st ` z ) , ( ( ( ( ( 1st ` z ) + ( 2nd ` z ) ) / 2 ) + ( 2nd ` z ) ) / 2 ) ) /\ ( 2nd ` ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) ) = if ( ( ( ( 1st ` z ) + ( 2nd ` z ) ) / 2 ) < w , ( ( ( 1st ` z ) + ( 2nd ` z ) ) / 2 ) , ( 2nd ` z ) ) ) )
30	29	simp1d 1073	. . . 4 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> ( <. ( 1st ` z ) , ( 2nd ` z ) >. D w ) e. ( RR X. RR ) )
31	19, 30	eqeltrd 2701	. . 3 |- ( ( ph /\ ( z e. ( RR X. RR ) /\ w e. RR ) ) -> ( z D w ) e. ( RR X. RR ) )
32		nn0uz 11722	. . 3 |- NN0 = ( ZZ>= ` 0 )
33		0zd 11389	. . 3 |- ( ph -> 0 e. ZZ )
34		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
35	34	fveq2i 6194	. . . . . 6 |- ( ZZ>= ` ( 0 + 1 ) ) = ( ZZ>= ` 1 )
36		nnuz 11723	. . . . . 6 |- NN = ( ZZ>= ` 1 )
37	35, 36	eqtr4i 2647	. . . . 5 |- ( ZZ>= ` ( 0 + 1 ) ) = NN
38	37	eleq2i 2693	. . . 4 |- ( z e. ( ZZ>= ` ( 0 + 1 ) ) <-> z e. NN )
39	9	equncomi 3759	. . . . . . . 8 |- C = ( F u. { <. 0 , <. 0 , 1 >. >. } )
40	39	fveq1i 6192	. . . . . . 7 |- ( C ` z ) = ( ( F u. { <. 0 , <. 0 , 1 >. >. } ) ` z )
41		nnne0 11053	. . . . . . . . 9 |- ( z e. NN -> z =/= 0 )
42	41	necomd 2849	. . . . . . . 8 |- ( z e. NN -> 0 =/= z )
43		fvunsn 6445	. . . . . . . 8 |- ( 0 =/= z -> ( ( F u. { <. 0 , <. 0 , 1 >. >. } ) ` z ) = ( F ` z ) )
44	42, 43	syl 17	. . . . . . 7 |- ( z e. NN -> ( ( F u. { <. 0 , <. 0 , 1 >. >. } ) ` z ) = ( F ` z ) )
45	40, 44	syl5eq 2668	. . . . . 6 |- ( z e. NN -> ( C ` z ) = ( F ` z ) )
46	45	adantl 482	. . . . 5 |- ( ( ph /\ z e. NN ) -> ( C ` z ) = ( F ` z ) )
47	7	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ z e. NN ) -> ( F ` z ) e. RR )
48	46, 47	eqeltrd 2701	. . . 4 |- ( ( ph /\ z e. NN ) -> ( C ` z ) e. RR )
49	38, 48	sylan2b 492	. . 3 |- ( ( ph /\ z e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( C ` z ) e. RR )
50	16, 31, 32, 33, 49	seqf2 12820	. 2 |- ( ph -> seq 0 ( D , C ) : NN0 --> ( RR X. RR ) )
51	1	feq1i 6036	. 2 |- ( G : NN0 --> ( RR X. RR ) <-> seq 0 ( D , C ) : NN0 --> ( RR X. RR ) )
52	50, 51	sylibr 224	1 |- ( ph -> G : NN0 --> ( RR X. RR ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   [_csb 3533   u. cun 3572   ifcif 4086   {csn 4177   <.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   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801

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

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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  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-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  ruclem8  14966  ruclem9  14967  ruclem10  14968  ruclem11  14969  ruclem12  14970