Theorem ruclem11 14969

Description: Lemma for ruc 14972. Closure lemmas for supremum. (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
ruclem11	|- ( ph -> ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ A. z e. ran ( 1st o. G ) z <_ 1 ) )

Distinct variable groups:   x, m, y   z, C   z, m, F, x, y   m, G, x, y, z   ph, z   z, D

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

Proof of Theorem ruclem11

Dummy variable n is distinct from all other variables.

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		1stcof 7196	. . . 4 |- ( G : NN0 --> ( RR X. RR ) -> ( 1st o. G ) : NN0 --> RR )
7	5, 6	syl 17	. . 3 |- ( ph -> ( 1st o. G ) : NN0 --> RR )
8		frn 6053	. . 3 |- ( ( 1st o. G ) : NN0 --> RR -> ran ( 1st o. G ) C_ RR )
9	7, 8	syl 17	. 2 |- ( ph -> ran ( 1st o. G ) C_ RR )
10		fdm 6051	. . . . 5 |- ( ( 1st o. G ) : NN0 --> RR -> dom ( 1st o. G ) = NN0 )
11	7, 10	syl 17	. . . 4 |- ( ph -> dom ( 1st o. G ) = NN0 )
12		0nn0 11307	. . . . 5 |- 0 e. NN0
13		ne0i 3921	. . . . 5 |- ( 0 e. NN0 -> NN0 =/= (/) )
14	12, 13	mp1i 13	. . . 4 |- ( ph -> NN0 =/= (/) )
15	11, 14	eqnetrd 2861	. . 3 |- ( ph -> dom ( 1st o. G ) =/= (/) )
16		dm0rn0 5342	. . . 4 |- ( dom ( 1st o. G ) = (/) <-> ran ( 1st o. G ) = (/) )
17	16	necon3bii 2846	. . 3 |- ( dom ( 1st o. G ) =/= (/) <-> ran ( 1st o. G ) =/= (/) )
18	15, 17	sylib 208	. 2 |- ( ph -> ran ( 1st o. G ) =/= (/) )
19		fvco3 6275	. . . . . 6 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
20	5, 19	sylan 488	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
21	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> F : NN --> RR )
22	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> 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		simpr 477	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> n e. NN0 )
24	12	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> 0 e. NN0 )
25	21, 22, 3, 4, 23, 24	ruclem10 14968	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> ( 1st ` ( G ` n ) ) < ( 2nd ` ( G ` 0 ) ) )
26	1, 2, 3, 4	ruclem4 14963	. . . . . . . . . 10 |- ( ph -> ( G ` 0 ) = <. 0 , 1 >. )
27	26	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( 2nd ` ( G ` 0 ) ) = ( 2nd ` <. 0 , 1 >. ) )
28		c0ex 10034	. . . . . . . . . 10 |- 0 e. _V
29		1ex 10035	. . . . . . . . . 10 |- 1 e. _V
30	28, 29	op2nd 7177	. . . . . . . . 9 |- ( 2nd ` <. 0 , 1 >. ) = 1
31	27, 30	syl6eq 2672	. . . . . . . 8 |- ( ph -> ( 2nd ` ( G ` 0 ) ) = 1 )
32	31	adantr 481	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> ( 2nd ` ( G ` 0 ) ) = 1 )
33	25, 32	breqtrd 4679	. . . . . 6 |- ( ( ph /\ n e. NN0 ) -> ( 1st ` ( G ` n ) ) < 1 )
34	5	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> ( G ` n ) e. ( RR X. RR ) )
35		xp1st 7198	. . . . . . . 8 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 1st ` ( G ` n ) ) e. RR )
36	34, 35	syl 17	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> ( 1st ` ( G ` n ) ) e. RR )
37		1re 10039	. . . . . . 7 |- 1 e. RR
38		ltle 10126	. . . . . . 7 |- ( ( ( 1st ` ( G ` n ) ) e. RR /\ 1 e. RR ) -> ( ( 1st ` ( G ` n ) ) < 1 -> ( 1st ` ( G ` n ) ) <_ 1 ) )
39	36, 37, 38	sylancl 694	. . . . . 6 |- ( ( ph /\ n e. NN0 ) -> ( ( 1st ` ( G ` n ) ) < 1 -> ( 1st ` ( G ` n ) ) <_ 1 ) )
40	33, 39	mpd 15	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( 1st ` ( G ` n ) ) <_ 1 )
41	20, 40	eqbrtrd 4675	. . . 4 |- ( ( ph /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) <_ 1 )
42	41	ralrimiva 2966	. . 3 |- ( ph -> A. n e. NN0 ( ( 1st o. G ) ` n ) <_ 1 )
43		ffn 6045	. . . . 5 |- ( ( 1st o. G ) : NN0 --> RR -> ( 1st o. G ) Fn NN0 )
44	7, 43	syl 17	. . . 4 |- ( ph -> ( 1st o. G ) Fn NN0 )
45		breq1 4656	. . . . 5 |- ( z = ( ( 1st o. G ) ` n ) -> ( z <_ 1 <-> ( ( 1st o. G ) ` n ) <_ 1 ) )
46	45	ralrn 6362	. . . 4 |- ( ( 1st o. G ) Fn NN0 -> ( A. z e. ran ( 1st o. G ) z <_ 1 <-> A. n e. NN0 ( ( 1st o. G ) ` n ) <_ 1 ) )
47	44, 46	syl 17	. . 3 |- ( ph -> ( A. z e. ran ( 1st o. G ) z <_ 1 <-> A. n e. NN0 ( ( 1st o. G ) ` n ) <_ 1 ) )
48	42, 47	mpbird 247	. 2 |- ( ph -> A. z e. ran ( 1st o. G ) z <_ 1 )
49	9, 18, 48	3jca 1242	1 |- ( ph -> ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ A. z e. ran ( 1st o. G ) z <_ 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->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:  ruclem12  14970