Theorem ruclem35 7495

Description: Lemma for ruc 7500. The supremum we have constructed lies between all values of the G and H functions. Compare ruclem29 7489, 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 7486	. . 3 |- (G` A) < (G` (A + 1))
8	1, 2, 3, 4, 5	ruclem17 7477	. . . . . . 7 |- G:NN-->RR
9		ffn 3619	. . . . . . 7 |- (G:NN-->RR -> G Fn NN)
10	8, 9	ax-mp 7	. . . . . 6 |- G Fn NN
11		peano2nn 5891	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
12	6, 11	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
13		fnfvelrn 3804	. . . . . 6 |- ((G Fn NN /\ (A + 1) e. NN) -> (G` (A + 1)) e. ran G)
14	10, 12, 13	mp2an 696	. . . . 5 |- (G` (A + 1)) e. ran G
15	1, 2, 3, 4, 5	ruclem33 7493	. . . . . 6 |- (ran G (_ RR /\ ran G =/= (/) /\ E.w e. RR A.v e. ran G v <_ w)
16	15	suprubi 6017	. . . . 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 2635	. . 3 |- (G` (A + 1)) <_ S
20	1, 2, 3, 4, 5, 6	ruclem22 7482	. . . 4 |- (G` A) e. RR
21	1, 2, 3, 4, 5, 12	ruclem22 7482	. . . 4 |- (G` (A + 1)) e. RR
22	1, 2, 3, 4, 5, 18	ruclem34 7494	. . . 4 |- S e. RR
23	20, 21, 22	ltletr 5569	. . 3 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) <_ S) -> (G` A) < S)
24	7, 19, 23	mp2an 696	. 2 |- (G` A) < S
25	1, 2, 3, 4, 5, 12	ruclem23 7483	. . . . . 6 |- (H` (A + 1)) e. RR
26		fvelrnb 3751	. . . . . . . . 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 2618	. . . . . . . . . . 11 |- ((G` w) = u -> ((H` (A + 1)) < (G` w) <-> (H` (A + 1)) < u))
29	28	negbid 610	. . . . . . . . . 10 |- ((G` w) = u -> (-. (H` (A + 1)) < (G` w) <-> -. (H` (A + 1)) < u))
30		ltnsymt 5513	. . . . . . . . . . 11 |- (((G` w) e. RR /\ (H` (A + 1)) e. RR) -> ((G` w) < (H` (A + 1)) -> -. (H` (A + 1)) < (G` w)))
31		fveq2 3715	. . . . . . . . . . . . . 14 |- (w = if(w e. NN, w, 1) -> (G` w) = (G` if(w e. NN, w, 1)))
32	31	eleq1d 1537	. . . . . . . . . . . . 13 |- (w = if(w e. NN, w, 1) -> ((G` w) e. RR <-> (G` if(w e. NN, w, 1)) e. RR))
33		1nn 5890	. . . . . . . . . . . . . . 15 |- 1 e. NN
34	33	elimel 2390	. . . . . . . . . . . . . 14 |- if(w e. NN, w, 1) e. NN
35	1, 2, 3, 4, 5, 34	ruclem22 7482	. . . . . . . . . . . . 13 |- (G` if(w e. NN, w, 1)) e. RR
36	32, 35	dedth 2379	. . . . . . . . . . . 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 2624	. . . . . . . . . . . 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 7492	. . . . . . . . . . . 12 |- (G` if(w e. NN, w, 1)) < (H` (A + 1))
40	38, 39	dedth 2379	. . . . . . . . . . 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 1740	. . . . . . . 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 1695	. . . . . 6 |- A.u e. ran G -. (H` (A + 1)) < u
46	15	suprnubi 6019	. . . . . 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 696	. . . . 5 |- -. (H` (A + 1)) < sup(ran G, RR, < )
48	18	breq2i 2622	. . . . 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 5559	. . . 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 7487	. . 3 |- (H` (A + 1)) < (H` A)
53	1, 2, 3, 4, 5, 6	ruclem23 7483	. . . 4 |- (H` A) e. RR
54	22, 25, 53	lelttr 5568	. . 3 |- ((S <_ (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> S < (H` A))
55	51, 52, 54	mp2an 696	. 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 954   e. wcel 956  A.wral 1642  E.wrex 1643   \ cdif 2040   u. cun 2041  ifcif 2357  {csn 2405  <.cop 2407   class class class wbr 2614   X. cxp 3163  ran crn 3166   |` cres 3167   o. ccom 3169   Fn wfn 3172  -->wf 3173  ` cfv 3177  (class class class)co 3954  {copab2 3955  1stc1st 4067  2ndc2nd 4068  supcsup 4553  RRcr 5213  1c1 5215   + caddc 5217   x. cmul 5219   / cdiv 5274   <_ cle 5275  NNcn 5276   < clt 5466  2c2 5916  3c3 5917   seq1 cseq1 6252

This theorem is referenced by:  ruclem36 7496

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-sup 4554  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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