Theorem ruclem26 7486

Description: Lemma for ruc 7500. Our constructed function G has an ever-increasing set of values.

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))
ruclem18.a	|- A e. NN

Assertion

Ref	Expression
ruclem26	|- (G` A) < (G` (A + 1))

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

Proof of Theorem ruclem26

Step	Hyp	Ref	Expression
1		ruclem.0	. . . . . 6 |- F:NN-->RR
2		ruclem.1	. . . . . 6 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
3		ruclem.2	. . . . . 6 |- 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)>.))}
4		ruclem.3	. . . . . 6 |- G = (1st o. (D seq1 C))
5		ruclem.4	. . . . . 6 |- H = (2nd o. (D seq1 C))
6		ruclem18.a	. . . . . 6 |- A e. NN
7	1, 2, 3, 4, 5, 6	ruclem22 7482	. . . . 5 |- (G` A) e. RR
8		peano2nn 5891	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
9	6, 8	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
10		ffvelrn 3805	. . . . . 6 |- ((F:NN-->RR /\ (A + 1) e. NN) -> (F` (A + 1)) e. RR)
11	1, 9, 10	mp2an 696	. . . . 5 |- (F` (A + 1)) e. RR
12		2re 5934	. . . . . . . 8 |- 2 e. RR
13	12, 11	remulcl 5315	. . . . . . 7 |- (2 x. (F` (A + 1))) e. RR
14	1, 2, 3, 4, 5, 6	ruclem23 7483	. . . . . . 7 |- (H` A) e. RR
15	13, 14	readdcl 5314	. . . . . 6 |- ((2 x. (F` (A + 1))) + (H` A)) e. RR
16		3re 5936	. . . . . 6 |- 3 e. RR
17		3pos 5946	. . . . . . 7 |- 0 < 3
18	16, 17	gt0ne0i 5599	. . . . . 6 |- 3 =/= 0
19	15, 16, 18	redivcl 5762	. . . . 5 |- (((2 x. (F` (A + 1))) + (H` A)) / 3) e. RR
20	7, 11, 19	lttr 5567	. . . 4 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3)) -> (G` A) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
21	11, 14	ruclem1 7461	. . . 4 |- ((F` (A + 1)) < (H` A) <-> (F` (A + 1)) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
22	20, 21	sylan2b 452	. . 3 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` A) < (((2 x. (F` (A + 1))) + (H` A)) / 3))
23	1, 2, 3, 4, 5, 6	ruclem18 7478	. . 3 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` (A + 1)) = (((2 x. (F` (A + 1))) + (H` A)) / 3))
24	22, 23	breqtrrd 2636	. 2 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` A) < (G` (A + 1)))
25	1, 2, 3, 4, 5, 6	ruclem20 7480	. . 3 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` (A + 1)) = (((2 x. (G` A)) + (H` A)) / 3))
26	1, 2, 3, 4, 5, 6	ruclem25 7485	. . . 4 |- (G` A) < (H` A)
27	7, 14	ruclem1 7461	. . . 4 |- ((G` A) < (H` A) <-> (G` A) < (((2 x. (G` A)) + (H` A)) / 3))
28	26, 27	mpbi 189	. . 3 |- (G` A) < (((2 x. (G` A)) + (H` A)) / 3)
29	25, 28	syl5breqr 2646	. 2 |- (-. ((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` A) < (G` (A + 1)))
30	24, 29	pm2.61i 126	1 |- (G` A) < (G` (A + 1))

Syntax hints:  -. wn 2   /\ wa 223   = wceq 954   e. wcel 956   \ cdif 2040   u. cun 2041  ifcif 2357  {csn 2405  <.cop 2407   class class class wbr 2614   X. cxp 3163   |` cres 3167   o. ccom 3169  -->wf 3173  ` cfv 3177  (class class class)co 3954  {copab2 3955  1stc1st 4067  2ndc2nd 4068  RRcr 5213  1c1 5215   + caddc 5217   x. cmul 5219   / cdiv 5274  NNcn 5276   < clt 5466  2c2 5916  3c3 5917   seq1 cseq1 6252

This theorem is referenced by:  ruclem28 7488  ruclem30 7490  ruclem32 7492  ruclem35 7495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-2 5925  df-3 5926  df-n0 6055  df-z 6091  df-seq1 6253

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