Theorem ruclem28 7497

Description: Lemma for ruc 7509. A helper lemma for ruclem29 7498.

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 5893	. . . . . . . . . 10 |- (A e. NN -> (A + 1) e. NN)
4	2, 3	ax-mp 7	. . . . . . . . 9 |- (A + 1) e. NN
5		ffvelrn 3809	. . . . . . . . 9 |- ((F:NN-->RR /\ (A + 1) e. NN) -> (F` (A + 1)) e. RR)
6	1, 4, 5	mp2an 696	. . . . . . . 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 7492	. . . . . . . 8 |- (H` A) e. RR
12	6, 11	ruclem1 7470	. . . . . . 7 |- ((F` (A + 1)) < (H` A) <-> (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
13	12	biimp 151	. . . . . 6 |- ((F` (A + 1)) < (H` A) -> (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
14	13	adantl 388	. . . . 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 7487	. . . . 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 2637	. . . 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 7491	. . . . 5 |- (G` (A + 1)) e. RR
18	6, 17	ltnsym 5560	. . . 4 |- ((F` (A + 1)) < (G` (A + 1)) -> -. (G` (A + 1)) < (F` (A + 1)))
19	16, 18	syl 10	. . 3 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> -. (G` (A + 1)) < (F` (A + 1)))
20	19	intnanrd 693	. 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 7495	. . . . 5 |- (G` A) < (G` (A + 1))
22	1, 7, 8, 9, 10, 2	ruclem22 7491	. . . . . 6 |- (G` A) e. RR
23	22, 17, 6	lttr 5569	. . . . 5 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) < (F` (A + 1))) -> (G` A) < (F` (A + 1)))
24	21, 23	mpan 694	. . . 4 |- ((G` (A + 1)) < (F` (A + 1)) -> (G` A) < (F` (A + 1)))
25	1, 7, 8, 9, 10, 2	ruclem27 7496	. . . . 5 |- (H` (A + 1)) < (H` A)
26	1, 7, 8, 9, 10, 4	ruclem23 7492	. . . . . 6 |- (H` (A + 1)) e. RR
27	6, 26, 11	lttr 5569	. . . . 5 |- (((F` (A + 1)) < (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> (F` (A + 1)) < (H` A))
28	25, 27	mpan2 695	. . . 4 |- ((F` (A + 1)) < (H` (A + 1)) -> (F` (A + 1)) < (H` A))
29	24, 28	anim12i 333	. . 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 98	. 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 126	1 |- -. ((G` (A + 1)) < (F` (A + 1)) /\ (F` (A + 1)) < (H` (A + 1)))

Syntax hints:  -. wn 2   /\ wa 223   = wceq 955   e. wcel 957   \ cdif 2041   u. cun 2042  ifcif 2358  {csn 2406  <.cop 2408   class class class wbr 2615   X. cxp 3164   |` cres 3168   o. ccom 3170  -->wf 3174  ` cfv 3178  (class class class)co 3958  {copab2 3959  1stc1st 4070  2ndc2nd 4071  RRcr 5216  1c1 5218   + caddc 5220   x. cmul 5222   / cdiv 5277  NNcn 5279   < clt 5469  2c2 5918  3c3 5919   seq1 cseq1 6257

This theorem is referenced by:  ruclem29 7498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-n 5883  df-2 5927  df-3 5928  df-n0 6057  df-z 6093  df-seq1 6258

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