Theorem ruclem26 7535

Description: Lemma for ruc 7549. 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 7531	. . . . 5 |- (G` A) e. RR
8		peano2nn 5935	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
9	6, 8	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
10		ffvelrn 3814	. . . . . 6 |- ((F:NN-->RR /\ (A + 1) e. NN) -> (F` (A + 1)) e. RR)
11	1, 9, 10	mp2an 697	. . . . 5 |- (F` (A + 1)) e. RR
12		2re 5979	. . . . . . . 8 |- 2 e. RR
13	12, 11	remulcl 5335	. . . . . . 7 |- (2 x. (F` (A + 1))) e. RR
14	1, 2, 3, 4, 5, 6	ruclem23 7532	. . . . . . 7 |- (H` A) e. RR
15	13, 14	readdcl 5334	. . . . . 6 |- ((2 x. (F` (A + 1))) + (H` A)) e. RR
16		3re 5981	. . . . . 6 |- 3 e. RR
17		3pos 5991	. . . . . . 7 |- 0 < 3
18	16, 17	gt0ne0i 5617	. . . . . 6 |- 3 =/= 0
19	15, 16, 18	redivcl 5798	. . . . 5 |- (((2 x. (F` (A + 1))) + (H` A)) / 3) e. RR
20	7, 11, 19	lttr 5585	. . . 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 7510	. . . 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 7527	. . 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 2641	. 2 |- (((G` A) < (F` (A + 1)) /\ (F` (A + 1)) < (H` A)) -> (G` A) < (G` (A + 1)))
25	1, 2, 3, 4, 5, 6	ruclem20 7529	. . 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 7534	. . . 4 |- (G` A) < (H` A)
27	7, 14	ruclem1 7510	. . . 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 2651	. 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 956   e. wcel 958   \ cdif 2044   u. cun 2045  ifcif 2361  {csn 2409  <.cop 2411   class class class wbr 2619   X. cxp 3168   |` cres 3172   o. ccom 3174  -->wf 3178  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  2ndc2nd 4078  RRcr 5233  1c1 5235   + caddc 5237   x. cmul 5239   / cdiv 5294  NNcn 5296   < clt 5486  2c2 5961  3c3 5962   seq1 cseq1 6307

This theorem is referenced by:  ruclem28 7537  ruclem30 7539  ruclem32 7541  ruclem35 7544

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-n0 6100  df-z 6136  df-seq1 6308

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