Theorem ruclem10 14968

Description: Lemma for ruc 14972. Every first component of the G sequence is less than every second component. That is, the sequences form a chain a₁ < a₂ <... < b₂ < b₁, where a_i are the first components and b_i are the second components. (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 )
ruclem10.6	|- ( ph -> M e. NN0 )
ruclem10.7	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
ruclem10	|- ( ph -> ( 1st ` ( G ` M ) ) < ( 2nd ` ( G ` N ) ) )

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

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

Proof of Theorem ruclem10

Step	Hyp	Ref	Expression
1		ruc.1	. . . . 5 |- ( ph -> F : NN --> RR )
2		ruc.2	. . . . 5 |- ( 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 ) >. ) ) )
3		ruc.4	. . . . 5 |- C = ( { <. 0 , <. 0 , 1 >. >. } u. F )
4		ruc.5	. . . . 5 |- G = seq 0 ( D , C )
5	1, 2, 3, 4	ruclem6 14964	. . . 4 |- ( ph -> G : NN0 --> ( RR X. RR ) )
6		ruclem10.6	. . . 4 |- ( ph -> M e. NN0 )
7	5, 6	ffvelrnd 6360	. . 3 |- ( ph -> ( G ` M ) e. ( RR X. RR ) )
8		xp1st 7198	. . 3 |- ( ( G ` M ) e. ( RR X. RR ) -> ( 1st ` ( G ` M ) ) e. RR )
9	7, 8	syl 17	. 2 |- ( ph -> ( 1st ` ( G ` M ) ) e. RR )
10		ruclem10.7	. . . . 5 |- ( ph -> N e. NN0 )
11	10, 6	ifcld 4131	. . . 4 |- ( ph -> if ( M <_ N , N , M ) e. NN0 )
12	5, 11	ffvelrnd 6360	. . 3 |- ( ph -> ( G ` if ( M <_ N , N , M ) ) e. ( RR X. RR ) )
13		xp1st 7198	. . 3 |- ( ( G ` if ( M <_ N , N , M ) ) e. ( RR X. RR ) -> ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) e. RR )
14	12, 13	syl 17	. 2 |- ( ph -> ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) e. RR )
15	5, 10	ffvelrnd 6360	. . 3 |- ( ph -> ( G ` N ) e. ( RR X. RR ) )
16		xp2nd 7199	. . 3 |- ( ( G ` N ) e. ( RR X. RR ) -> ( 2nd ` ( G ` N ) ) e. RR )
17	15, 16	syl 17	. 2 |- ( ph -> ( 2nd ` ( G ` N ) ) e. RR )
18	6	nn0red 11352	. . . . . 6 |- ( ph -> M e. RR )
19	10	nn0red 11352	. . . . . 6 |- ( ph -> N e. RR )
20		max1 12016	. . . . . 6 |- ( ( M e. RR /\ N e. RR ) -> M <_ if ( M <_ N , N , M ) )
21	18, 19, 20	syl2anc 693	. . . . 5 |- ( ph -> M <_ if ( M <_ N , N , M ) )
22	6	nn0zd 11480	. . . . . 6 |- ( ph -> M e. ZZ )
23	11	nn0zd 11480	. . . . . 6 |- ( ph -> if ( M <_ N , N , M ) e. ZZ )
24		eluz 11701	. . . . . 6 |- ( ( M e. ZZ /\ if ( M <_ N , N , M ) e. ZZ ) -> ( if ( M <_ N , N , M ) e. ( ZZ>= ` M ) <-> M <_ if ( M <_ N , N , M ) ) )
25	22, 23, 24	syl2anc 693	. . . . 5 |- ( ph -> ( if ( M <_ N , N , M ) e. ( ZZ>= ` M ) <-> M <_ if ( M <_ N , N , M ) ) )
26	21, 25	mpbird 247	. . . 4 |- ( ph -> if ( M <_ N , N , M ) e. ( ZZ>= ` M ) )
27	1, 2, 3, 4, 6, 26	ruclem9 14967	. . 3 |- ( ph -> ( ( 1st ` ( G ` M ) ) <_ ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) /\ ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) <_ ( 2nd ` ( G ` M ) ) ) )
28	27	simpld 475	. 2 |- ( ph -> ( 1st ` ( G ` M ) ) <_ ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) )
29		xp2nd 7199	. . . 4 |- ( ( G ` if ( M <_ N , N , M ) ) e. ( RR X. RR ) -> ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) e. RR )
30	12, 29	syl 17	. . 3 |- ( ph -> ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) e. RR )
31	1, 2, 3, 4	ruclem8 14966	. . . 4 |- ( ( ph /\ if ( M <_ N , N , M ) e. NN0 ) -> ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) < ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) )
32	11, 31	mpdan 702	. . 3 |- ( ph -> ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) < ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) )
33		max2 12018	. . . . . . 7 |- ( ( M e. RR /\ N e. RR ) -> N <_ if ( M <_ N , N , M ) )
34	18, 19, 33	syl2anc 693	. . . . . 6 |- ( ph -> N <_ if ( M <_ N , N , M ) )
35	10	nn0zd 11480	. . . . . . 7 |- ( ph -> N e. ZZ )
36		eluz 11701	. . . . . . 7 |- ( ( N e. ZZ /\ if ( M <_ N , N , M ) e. ZZ ) -> ( if ( M <_ N , N , M ) e. ( ZZ>= ` N ) <-> N <_ if ( M <_ N , N , M ) ) )
37	35, 23, 36	syl2anc 693	. . . . . 6 |- ( ph -> ( if ( M <_ N , N , M ) e. ( ZZ>= ` N ) <-> N <_ if ( M <_ N , N , M ) ) )
38	34, 37	mpbird 247	. . . . 5 |- ( ph -> if ( M <_ N , N , M ) e. ( ZZ>= ` N ) )
39	1, 2, 3, 4, 10, 38	ruclem9 14967	. . . 4 |- ( ph -> ( ( 1st ` ( G ` N ) ) <_ ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) /\ ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) <_ ( 2nd ` ( G ` N ) ) ) )
40	39	simprd 479	. . 3 |- ( ph -> ( 2nd ` ( G ` if ( M <_ N , N , M ) ) ) <_ ( 2nd ` ( G ` N ) ) )
41	14, 30, 17, 32, 40	ltletrd 10197	. 2 |- ( ph -> ( 1st ` ( G ` if ( M <_ N , N , M ) ) ) < ( 2nd ` ( G ` N ) ) )
42	9, 14, 17, 28, 41	lelttrd 10195	1 |- ( ph -> ( 1st ` ( G ` M ) ) < ( 2nd ` ( G ` N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = 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   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:  ruclem11  14969  ruclem12  14970