Theorem ruclem35 7487

Description: Lemma for ruc 7492. The supremum we have constructed lies between all values of the G and H functions. Compare ruclem29 7481, which states the opposite for the input function F.

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))
ruclem.5	|- S = sup(ran G, RR, < )
ruclem.a	|- A e. NN

Assertion

Ref	Expression
ruclem35	|- ((G` A) < S /\ S < (H` A))

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

Proof of Theorem ruclem35

Step	Hyp	Ref	Expression
1		ruclem.0	. . . 4 |- F:NN-->RR
2		ruclem.1	. . . 4 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
3		ruclem.2	. . . 4 |- 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	. . . 4 |- G = (1st o. (D seq1 C))
5		ruclem.4	. . . 4 |- H = (2nd o. (D seq1 C))
6		ruclem.a	. . . 4 |- A e. NN
7	1, 2, 3, 4, 5, 6	ruclem26 7478	. . 3 |- (G` A) < (G` (A + 1))
8	1, 2, 3, 4, 5	ruclem17 7469	. . . . . . 7 |- G:NN-->RR
9		ffn 3613	. . . . . . 7 |- (G:NN-->RR -> G Fn NN)
10	8, 9	ax-mp 7	. . . . . 6 |- G Fn NN
11		peano2nn 5883	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
12	6, 11	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
13		fnfvelrn 3798	. . . . . 6 |- ((G Fn NN /\ (A + 1) e. NN) -> (G` (A + 1)) e. ran G)
14	10, 12, 13	mp2an 695	. . . . 5 |- (G` (A + 1)) e. ran G
15	1, 2, 3, 4, 5	ruclem33 7485	. . . . . 6 |- (ran G (_ RR /\ ran G =/= (/) /\ E.w e. RR A.v e. ran G v <_ w)
16	15	suprubi 6009	. . . . 5 |- ((G` (A + 1)) e. ran G -> (G` (A + 1)) <_ sup(ran G, RR, < ))
17	14, 16	ax-mp 7	. . . 4 |- (G` (A + 1)) <_ sup(ran G, RR, < )
18		ruclem.5	. . . 4 |- S = sup(ran G, RR, < )
19	17, 18	breqtrr 2630	. . 3 |- (G` (A + 1)) <_ S
20	1, 2, 3, 4, 5, 6	ruclem22 7474	. . . 4 |- (G` A) e. RR
21	1, 2, 3, 4, 5, 12	ruclem22 7474	. . . 4 |- (G` (A + 1)) e. RR
22	1, 2, 3, 4, 5, 18	ruclem34 7486	. . . 4 |- S e. RR
23	20, 21, 22	ltletr 5561	. . 3 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) <_ S) -> (G` A) < S)
24	7, 19, 23	mp2an 695	. 2 |- (G` A) < S
25	1, 2, 3, 4, 5, 12	ruclem23 7475	. . . . . 6 |- (H` (A + 1)) e. RR
26		fvelrnb 3745	. . . . . . . . 9 |- (G Fn NN -> (u e. ran G <-> E.w e. NN (G` w) = u))
27	10, 26	ax-mp 7	. . . . . . . 8 |- (u e. ran G <-> E.w e. NN (G` w) = u)
28		breq2 2613	. . . . . . . . . . 11 |- ((G` w) = u -> ((H` (A + 1)) < (G` w) <-> (H` (A + 1)) < u))
29	28	negbid 609	. . . . . . . . . 10 |- ((G` w) = u -> (-. (H` (A + 1)) < (G` w) <-> -. (H` (A + 1)) < u))
30		ltnsymt 5505	. . . . . . . . . . 11 |- (((G` w) e. RR /\ (H` (A + 1)) e. RR) -> ((G` w) < (H` (A + 1)) -> -. (H` (A + 1)) < (G` w)))
31		fveq2 3709	. . . . . . . . . . . . . 14 |- (w = if(w e. NN, w, 1) -> (G` w) = (G` if(w e. NN, w, 1)))
32	31	eleq1d 1532	. . . . . . . . . . . . 13 |- (w = if(w e. NN, w, 1) -> ((G` w) e. RR <-> (G` if(w e. NN, w, 1)) e. RR))
33		1nn 5882	. . . . . . . . . . . . . . 15 |- 1 e. NN
34	33	elimel 2384	. . . . . . . . . . . . . 14 |- if(w e. NN, w, 1) e. NN
35	1, 2, 3, 4, 5, 34	ruclem22 7474	. . . . . . . . . . . . 13 |- (G` if(w e. NN, w, 1)) e. RR
36	32, 35	dedth 2373	. . . . . . . . . . . 12 |- (w e. NN -> (G` w) e. RR)
37	36, 25	jctir 293	. . . . . . . . . . 11 |- (w e. NN -> ((G` w) e. RR /\ (H` (A + 1)) e. RR))
38	31	breq1d 2619	. . . . . . . . . . . 12 |- (w = if(w e. NN, w, 1) -> ((G` w) < (H` (A + 1)) <-> (G` if(w e. NN, w, 1)) < (H` (A + 1))))
39	1, 2, 3, 4, 5, 34, 12	ruclem32 7484	. . . . . . . . . . . 12 |- (G` if(w e. NN, w, 1)) < (H` (A + 1))
40	38, 39	dedth 2373	. . . . . . . . . . 11 |- (w e. NN -> (G` w) < (H` (A + 1)))
41	30, 37, 40	sylc 68	. . . . . . . . . 10 |- (w e. NN -> -. (H` (A + 1)) < (G` w))
42	29, 41	syl5cbi 209	. . . . . . . . 9 |- (w e. NN -> ((G` w) = u -> -. (H` (A + 1)) < u))
43	42	r19.23aiv 1735	. . . . . . . 8 |- (E.w e. NN (G` w) = u -> -. (H` (A + 1)) < u)
44	27, 43	sylbi 199	. . . . . . 7 |- (u e. ran G -> -. (H` (A + 1)) < u)
45	44	rgen 1690	. . . . . 6 |- A.u e. ran G -. (H` (A + 1)) < u
46	15	suprnubi 6011	. . . . . 6 |- (((H` (A + 1)) e. RR /\ A.u e. ran G -. (H` (A + 1)) < u) -> -. (H` (A + 1)) < sup(ran G, RR, < ))
47	25, 45, 46	mp2an 695	. . . . 5 |- -. (H` (A + 1)) < sup(ran G, RR, < )
48	18	breq2i 2617	. . . . 5 |- ((H` (A + 1)) < S <-> (H` (A + 1)) < sup(ran G, RR, < ))
49	47, 48	mtbir 192	. . . 4 |- -. (H` (A + 1)) < S
50	22, 25	lenlt 5551	. . . 4 |- (S <_ (H` (A + 1)) <-> -. (H` (A + 1)) < S)
51	49, 50	mpbir 190	. . 3 |- S <_ (H` (A + 1))
52	1, 2, 3, 4, 5, 6	ruclem27 7479	. . 3 |- (H` (A + 1)) < (H` A)
53	1, 2, 3, 4, 5, 6	ruclem23 7475	. . . 4 |- (H` A) e. RR
54	22, 25, 53	lelttr 5560	. . 3 |- ((S <_ (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> S < (H` A))
55	51, 52, 54	mp2an 695	. 2 |- S < (H` A)
56	24, 55	pm3.2i 285	1 |- ((G` A) < S /\ S < (H` A))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638   \ cdif 2034   u. cun 2035  ifcif 2351  {csn 2399  <.cop 2401   class class class wbr 2609   X. cxp 3158  ran crn 3161   |` cres 3162   o. ccom 3164   Fn wfn 3167  -->wf 3168  ` cfv 3172  (class class class)co 3948  {copab2 3949  1stc1st 4061  2ndc2nd 4062  supcsup 4547  RRcr 5205  1c1 5207   + caddc 5209   x. cmul 5211   / cdiv 5266   <_ cle 5267  NNcn 5268   < clt 5458  2c2 5908  3c3 5909   seq1 cseq1 6244

This theorem is referenced by:  ruclem36 7488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-n0 6047  df-z 6083  df-seq1 6245

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