Theorem ruclem28 8806

Description: Lemma for ruc 8818. A helper lemma for ruclem29 8807.

Hypotheses

Ref	Expression
ruclem.0	|- F:NN-->RR
ruclem.1	|- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
ruclem.2	|- D = {<.<.x, y>., z>. | ((x e. (RR X. RR) /\ y e. RR) /\ z = if(((1st` x) < y /\ y < (2nd` x)), <.(((2 x. y) + (2nd` x)) / 3), ((y + (2 x. (2nd` x))) / 3)>., <.(((2 x. (1st` x)) + (2nd` x)) / 3), (((1st` x) + (2 x. (2nd` x))) / 3)>.))}
ruclem.3	|- G = (1st o. (D seq1 C))
ruclem.4	|- H = (2nd o. (D seq1 C))
ruclem28.a	|- A e. NN

Assertion

Ref	Expression
ruclem28	|- -. ((G` (A + 1)) < (F` (A + 1)) /\ (F` (A + 1)) < (H` (A + 1)))

Distinct variable groups:   x,y,z   z,F

Proof of Theorem ruclem28

Step	Hyp	Ref	Expression
1		ruclem.0	. . . . . . . . 9 |- F:NN-->RR
2		ruclem28.a	. . . . . . . . . 10 |- A e. NN
3		peano2nn 7118	. . . . . . . . . 10 |- (A e. NN -> (A + 1) e. NN)
4	2, 3	ax-mp 7	. . . . . . . . 9 |- (A + 1) e. NN
5		ffvelrn 4787	. . . . . . . . 9 |- ((F:NN-->RR /\ (A + 1) e. NN) -> (F` (A + 1)) e. RR)
6	1, 4, 5	mp2an 761	. . . . . . . 8 |- (F` (A + 1)) e. RR
7		ruclem.1	. . . . . . . . 9 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
8		ruclem.2	. . . . . . . . 9 |- D = {<.<.x, y>., z>. | ((x e. (RR X. RR) /\ y e. RR) /\ z = if(((1st` x) < y /\ y < (2nd` x)), <.(((2 x. y) + (2nd` x)) / 3), ((y + (2 x. (2nd` x))) / 3)>., <.(((2 x. (1st` x)) + (2nd` x)) / 3), (((1st` x) + (2 x. (2nd` x))) / 3)>.))}
9		ruclem.3	. . . . . . . . 9 |- G = (1st o. (D seq1 C))
10		ruclem.4	. . . . . . . . 9 |- H = (2nd o. (D seq1 C))
11	1, 7, 8, 9, 10, 2	ruclem23 8801	. . . . . . . 8 |- (H` A) e. RR
12	6, 11	ruclem1 8779	. . . . . . 7 |- ((F` (A + 1)) < (H` A) <-> (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
13	12	biimpi 168	. . . . . 6 |- ((F` (A + 1)) < (H` A) -> (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
14	13	adantl 424	. . . . 5 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
15	1, 7, 8, 9, 10, 2	ruclem18 8796	. . . . 5 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` (A + 1)) = (((2 x. (F` (A + 1))) + (H` A)) / 3))
16	14, 15	breqtrrd 3363	. . . 4 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (F` (A + 1)) < (G` (A + 1)))
17	1, 7, 8, 9, 10, 4	ruclem22 8800	. . . . 5 |- (G` (A + 1)) e. RR
18	6, 17	ltnsymi 6752	. . . 4 |- ((F` (A + 1)) < (G` (A + 1)) -> -. (G` (A + 1)) < (F` (A + 1)))
19	16, 18	syl 12	. . 3 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> -. (G` (A + 1)) < (F` (A + 1)))
20	19	intnanrd 758	. 2 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> -. ((G` (A + 1)) < (F` (A + 1)) /\ (F` (A + 1)) < (H` (A + 1))))
21	1, 7, 8, 9, 10, 2	ruclem26 8804	. . . . 5 |- (G` A) < (G` (A + 1))
22	1, 7, 8, 9, 10, 2	ruclem22 8800	. . . . . 6 |- (G` A) e. RR
23	22, 17, 6	lttri 6760	. . . . 5 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) < (F` (A + 1))) -> (G` A) < (F` (A + 1)))
24	21, 23	mpan 759	. . . 4 |- ((G` (A + 1)) < (F` (A + 1)) -> (G` A) < (F` (A + 1)))
25	1, 7, 8, 9, 10, 2	ruclem27 8805	. . . . 5 |- (H` (A + 1)) < (H` A)
26	1, 7, 8, 9, 10, 4	ruclem23 8801	. . . . . 6 |- (H` (A + 1)) e. RR
27	6, 26, 11	lttri 6760	. . . . 5 |- (((F` (A + 1)) < (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> (F` (A + 1)) < (H` A))
28	25, 27	mpan2 760	. . . 4 |- ((F` (A + 1)) < (H` (A + 1)) -> (F` (A + 1)) < (H` A))
29	24, 28	anim12i 360	. . 3 |- (((G` (A + 1)) < (F` (A + 1)) /\ (F` (A + 1)) < (H` (A + 1))) -> ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)))
30	29	con3i 114	. 2 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> -. ((G` (A + 1)) < (F` (A + 1)) /\ (F` (A + 1)) < (H` (A + 1))))
31	20, 30	pm2.61i 140	1 |- -. ((G` (A + 1)) < (F` (A + 1)) /\ (F` (A + 1)) < (H` (A + 1)))

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   u. cun 2591  ifcif 2982  {csn 3044  <.cop 3046   class class class wbr 3338   X. cxp 3984   |` cres 3988   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  RRcr 6385  1c1 6387   + caddc 6389   x. cmul 6391   / cdiv 6447  NNcn 6449   < clt 6653  2c2 7145  3c3 7146   seq1 cseq1 7720

This theorem is referenced by:  ruclem29 8807

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-n0 7309  df-z 7345  df-seq1 7721

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