Theorem ruclem8 14966

Description: Lemma for ruc 14972. The intervals of the G sequence are all nonempty. (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
ruclem8	|- ( ( ph /\ N e. NN0 ) -> ( 1st ` ( G ` N ) ) < ( 2nd ` ( G ` N ) ) )

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

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

Proof of Theorem ruclem8

Dummy variables n k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( k = 0 -> ( G ` k ) = ( G ` 0 ) )
2	1	fveq2d 6195	. . . . 5 |- ( k = 0 -> ( 1st ` ( G ` k ) ) = ( 1st ` ( G ` 0 ) ) )
3	1	fveq2d 6195	. . . . 5 |- ( k = 0 -> ( 2nd ` ( G ` k ) ) = ( 2nd ` ( G ` 0 ) ) )
4	2, 3	breq12d 4666	. . . 4 |- ( k = 0 -> ( ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) <-> ( 1st ` ( G ` 0 ) ) < ( 2nd ` ( G ` 0 ) ) ) )
5	4	imbi2d 330	. . 3 |- ( k = 0 -> ( ( ph -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) ) <-> ( ph -> ( 1st ` ( G ` 0 ) ) < ( 2nd ` ( G ` 0 ) ) ) ) )
6		fveq2 6191	. . . . . 6 |- ( k = n -> ( G ` k ) = ( G ` n ) )
7	6	fveq2d 6195	. . . . 5 |- ( k = n -> ( 1st ` ( G ` k ) ) = ( 1st ` ( G ` n ) ) )
8	6	fveq2d 6195	. . . . 5 |- ( k = n -> ( 2nd ` ( G ` k ) ) = ( 2nd ` ( G ` n ) ) )
9	7, 8	breq12d 4666	. . . 4 |- ( k = n -> ( ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) <-> ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) )
10	9	imbi2d 330	. . 3 |- ( k = n -> ( ( ph -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) ) <-> ( ph -> ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) )
11		fveq2 6191	. . . . . 6 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
12	11	fveq2d 6195	. . . . 5 |- ( k = ( n + 1 ) -> ( 1st ` ( G ` k ) ) = ( 1st ` ( G ` ( n + 1 ) ) ) )
13	11	fveq2d 6195	. . . . 5 |- ( k = ( n + 1 ) -> ( 2nd ` ( G ` k ) ) = ( 2nd ` ( G ` ( n + 1 ) ) ) )
14	12, 13	breq12d 4666	. . . 4 |- ( k = ( n + 1 ) -> ( ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) <-> ( 1st ` ( G ` ( n + 1 ) ) ) < ( 2nd ` ( G ` ( n + 1 ) ) ) ) )
15	14	imbi2d 330	. . 3 |- ( k = ( n + 1 ) -> ( ( ph -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) ) <-> ( ph -> ( 1st ` ( G ` ( n + 1 ) ) ) < ( 2nd ` ( G ` ( n + 1 ) ) ) ) ) )
16		fveq2 6191	. . . . . 6 |- ( k = N -> ( G ` k ) = ( G ` N ) )
17	16	fveq2d 6195	. . . . 5 |- ( k = N -> ( 1st ` ( G ` k ) ) = ( 1st ` ( G ` N ) ) )
18	16	fveq2d 6195	. . . . 5 |- ( k = N -> ( 2nd ` ( G ` k ) ) = ( 2nd ` ( G ` N ) ) )
19	17, 18	breq12d 4666	. . . 4 |- ( k = N -> ( ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) <-> ( 1st ` ( G ` N ) ) < ( 2nd ` ( G ` N ) ) ) )
20	19	imbi2d 330	. . 3 |- ( k = N -> ( ( ph -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` k ) ) ) <-> ( ph -> ( 1st ` ( G ` N ) ) < ( 2nd ` ( G ` N ) ) ) ) )
21		0lt1 10550	. . . . 5 |- 0 < 1
22	21	a1i 11	. . . 4 |- ( ph -> 0 < 1 )
23		ruc.1	. . . . . . 7 |- ( ph -> F : NN --> RR )
24		ruc.2	. . . . . . 7 |- ( 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 ) >. ) ) )
25		ruc.4	. . . . . . 7 |- C = ( { <. 0 , <. 0 , 1 >. >. } u. F )
26		ruc.5	. . . . . . 7 |- G = seq 0 ( D , C )
27	23, 24, 25, 26	ruclem4 14963	. . . . . 6 |- ( ph -> ( G ` 0 ) = <. 0 , 1 >. )
28	27	fveq2d 6195	. . . . 5 |- ( ph -> ( 1st ` ( G ` 0 ) ) = ( 1st ` <. 0 , 1 >. ) )
29		c0ex 10034	. . . . . 6 |- 0 e. _V
30		1ex 10035	. . . . . 6 |- 1 e. _V
31	29, 30	op1st 7176	. . . . 5 |- ( 1st ` <. 0 , 1 >. ) = 0
32	28, 31	syl6eq 2672	. . . 4 |- ( ph -> ( 1st ` ( G ` 0 ) ) = 0 )
33	27	fveq2d 6195	. . . . 5 |- ( ph -> ( 2nd ` ( G ` 0 ) ) = ( 2nd ` <. 0 , 1 >. ) )
34	29, 30	op2nd 7177	. . . . 5 |- ( 2nd ` <. 0 , 1 >. ) = 1
35	33, 34	syl6eq 2672	. . . 4 |- ( ph -> ( 2nd ` ( G ` 0 ) ) = 1 )
36	22, 32, 35	3brtr4d 4685	. . 3 |- ( ph -> ( 1st ` ( G ` 0 ) ) < ( 2nd ` ( G ` 0 ) ) )
37	23	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> F : NN --> RR )
38	24	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> 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 ) >. ) ) )
39	23, 24, 25, 26	ruclem6 14964	. . . . . . . . . . . 12 |- ( ph -> G : NN0 --> ( RR X. RR ) )
40	39	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN0 ) -> ( G ` n ) e. ( RR X. RR ) )
41	40	adantrr 753	. . . . . . . . . 10 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( G ` n ) e. ( RR X. RR ) )
42		xp1st 7198	. . . . . . . . . 10 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 1st ` ( G ` n ) ) e. RR )
43	41, 42	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 1st ` ( G ` n ) ) e. RR )
44		xp2nd 7199	. . . . . . . . . 10 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 2nd ` ( G ` n ) ) e. RR )
45	41, 44	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 2nd ` ( G ` n ) ) e. RR )
46		nn0p1nn 11332	. . . . . . . . . . 11 |- ( n e. NN0 -> ( n + 1 ) e. NN )
47		ffvelrn 6357	. . . . . . . . . . 11 |- ( ( F : NN --> RR /\ ( n + 1 ) e. NN ) -> ( F ` ( n + 1 ) ) e. RR )
48	23, 46, 47	syl2an 494	. . . . . . . . . 10 |- ( ( ph /\ n e. NN0 ) -> ( F ` ( n + 1 ) ) e. RR )
49	48	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( F ` ( n + 1 ) ) e. RR )
50		eqid 2622	. . . . . . . . 9 |- ( 1st ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) = ( 1st ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) )
51		eqid 2622	. . . . . . . . 9 |- ( 2nd ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) = ( 2nd ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) )
52		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) )
53	37, 38, 43, 45, 49, 50, 51, 52	ruclem2 14961	. . . . . . . 8 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( ( 1st ` ( G ` n ) ) <_ ( 1st ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) /\ ( 1st ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) < ( 2nd ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) /\ ( 2nd ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) <_ ( 2nd ` ( G ` n ) ) ) )
54	53	simp2d 1074	. . . . . . 7 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 1st ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) < ( 2nd ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) )
55	23, 24, 25, 26	ruclem7 14965	. . . . . . . . . 10 |- ( ( ph /\ n e. NN0 ) -> ( G ` ( n + 1 ) ) = ( ( G ` n ) D ( F ` ( n + 1 ) ) ) )
56	55	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( G ` ( n + 1 ) ) = ( ( G ` n ) D ( F ` ( n + 1 ) ) ) )
57		1st2nd2 7205	. . . . . . . . . . 11 |- ( ( G ` n ) e. ( RR X. RR ) -> ( G ` n ) = <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. )
58	41, 57	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( G ` n ) = <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. )
59	58	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( ( G ` n ) D ( F ` ( n + 1 ) ) ) = ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) )
60	56, 59	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( G ` ( n + 1 ) ) = ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) )
61	60	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 1st ` ( G ` ( n + 1 ) ) ) = ( 1st ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) )
62	60	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 2nd ` ( G ` ( n + 1 ) ) ) = ( 2nd ` ( <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. D ( F ` ( n + 1 ) ) ) ) )
63	54, 61, 62	3brtr4d 4685	. . . . . 6 |- ( ( ph /\ ( n e. NN0 /\ ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) ) -> ( 1st ` ( G ` ( n + 1 ) ) ) < ( 2nd ` ( G ` ( n + 1 ) ) ) )
64	63	expr 643	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) -> ( 1st ` ( G ` ( n + 1 ) ) ) < ( 2nd ` ( G ` ( n + 1 ) ) ) ) )
65	64	expcom 451	. . . 4 |- ( n e. NN0 -> ( ph -> ( ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) -> ( 1st ` ( G ` ( n + 1 ) ) ) < ( 2nd ` ( G ` ( n + 1 ) ) ) ) ) )
66	65	a2d 29	. . 3 |- ( n e. NN0 -> ( ( ph -> ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` n ) ) ) -> ( ph -> ( 1st ` ( G ` ( n + 1 ) ) ) < ( 2nd ` ( G ` ( n + 1 ) ) ) ) ) )
67	5, 10, 15, 20, 36, 66	nn0ind 11472	. 2 |- ( N e. NN0 -> ( ph -> ( 1st ` ( G ` N ) ) < ( 2nd ` ( G ` N ) ) ) )
68	67	impcom 446	1 |- ( ( ph /\ N e. NN0 ) -> ( 1st ` ( G ` N ) ) < ( 2nd ` ( G ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   [_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   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   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:  ruclem9  14967  ruclem10  14968  ruclem12  14970