Theorem isupivth 7290

Description: The intermediate value theorem, increasing case with supremum solution. (Contributed by Paul Chapman, 22-Jan-2008.)

Hypotheses

Ref	Expression
isupivth.1	|- A e. RR
isupivth.2	|- B e. RR
isupivth.3	|- U e. RR
isupivth.4	|- A < B
isupivth.5	|- (A[,]B) (_ D
isupivth.6	|- D (_ CC
isupivth.7	|- F e. (D-cn->CC)
isupivth.8	|- (x e. (A[,]B) -> (F` x) e. RR)
isupivth.9	|- S = {x e. (A[,]B) | (F` x) = U}
isupivth.10	|- ((F` A) < U /\ U < (F` B))
isupivth.11	|- C = sup(S, RR, < )

Assertion

Ref	Expression
isupivth	|- (C e. (A(,)B) /\ (F` C) = U)

Distinct variable groups:   x,A   x,B   x,F   x,U

Proof of Theorem isupivth

Step	Hyp	Ref	Expression
1		isupivth.11	. . . 4 |- C = sup(S, RR, < )
2		isupivth.1	. . . . 5 |- A e. RR
3		isupivth.2	. . . . 5 |- B e. RR
4		isupivth.3	. . . . 5 |- U e. RR
5		isupivth.4	. . . . 5 |- A < B
6		isupivth.10	. . . . . 6 |- ((F` A) < U /\ U < (F` B))
7	2, 3, 5	ltlei 5593	. . . . . . . . . 10 |- A <_ B
8		lbicc2t 6405	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
9	2, 3, 7, 8	mp3an 918	. . . . . . . . 9 |- A e. (A[,]B)
10		fvres 3740	. . . . . . . . 9 |- (A e. (A[,]B) -> ((F |` (A[,]B))` A) = (F` A))
11	9, 10	ax-mp 7	. . . . . . . 8 |- ((F |` (A[,]B))` A) = (F` A)
12	11	breq1i 2631	. . . . . . 7 |- (((F |` (A[,]B))` A) < U <-> (F` A) < U)
13		ubicc2t 6406	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
14	2, 3, 7, 13	mp3an 918	. . . . . . . . 9 |- B e. (A[,]B)
15		fvres 3740	. . . . . . . . 9 |- (B e. (A[,]B) -> ((F |` (A[,]B))` B) = (F` B))
16	14, 15	ax-mp 7	. . . . . . . 8 |- ((F |` (A[,]B))` B) = (F` B)
17	16	breq2i 2632	. . . . . . 7 |- (U < ((F |` (A[,]B))` B) <-> U < (F` B))
18	12, 17	anbi12i 484	. . . . . 6 |- ((((F |` (A[,]B))` A) < U /\ U < ((F |` (A[,]B))` B)) <-> ((F` A) < U /\ U < (F` B)))
19	6, 18	mpbir 190	. . . . 5 |- (((F |` (A[,]B))` A) < U /\ U < ((F |` (A[,]B))` B))
20		fvres 3740	. . . . . . . . 9 |- (c e. (A[,]B) -> ((F |` (A[,]B))` c) = (F` c))
21	20	breq1d 2634	. . . . . . . 8 |- (c e. (A[,]B) -> (((F |` (A[,]B))` c) <_ U <-> (F` c) <_ U))
22	21	rabbii 1808	. . . . . . 7 |- {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | (F` c) <_ U}
23		supeq1 4584	. . . . . . 7 |- ({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | (F` c) <_ U} -> sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ))
24	22, 23	ax-mp 7	. . . . . 6 |- sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )
25	24	eqcomi 1482	. . . . 5 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < )
26		eqid 1478	. . . . 5 |- {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}
27		isupivth.7	. . . . . . 7 |- F e. (D-cn->CC)
28		isupivth.6	. . . . . . . 8 |- D (_ CC
29		ssid 2083	. . . . . . . 8 |- CC (_ CC
30		isupivth.5	. . . . . . . 8 |- (A[,]B) (_ D
31		rescncf 7272	. . . . . . . 8 |- ((D (_ CC /\ CC (_ CC /\ (A[,]B) (_ D) -> (F e. (D-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->CC)))
32	28, 29, 30, 31	mp3an 918	. . . . . . 7 |- (F e. (D-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->CC))
33	27, 32	ax-mp 7	. . . . . 6 |- (F |` (A[,]B)) e. ((A[,]B)-cn->CC)
34	30, 28	sstri 2076	. . . . . . . 8 |- (A[,]B) (_ CC
35		axresscn 5280	. . . . . . . 8 |- RR (_ CC
36	34, 29, 35	3pm3.2i 820	. . . . . . 7 |- ((A[,]B) (_ CC /\ CC (_ CC /\ RR (_ CC)
37		fvres 3740	. . . . . . . . 9 |- (x e. (A[,]B) -> ((F |` (A[,]B))` x) = (F` x))
38		isupivth.8	. . . . . . . . 9 |- (x e. (A[,]B) -> (F` x) e. RR)
39	37, 38	eqeltrd 1551	. . . . . . . 8 |- (x e. (A[,]B) -> ((F |` (A[,]B))` x) e. RR)
40	39	rgen 1701	. . . . . . 7 |- A.x e. (A[,]B)((F |` (A[,]B))` x) e. RR
41		cncffvrn 7273	. . . . . . 7 |- ((((A[,]B) (_ CC /\ CC (_ CC /\ RR (_ CC) /\ A.x e. (A[,]B)((F |` (A[,]B))` x) e. RR) -> ((F |` (A[,]B)) e. ((A[,]B)-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->RR)))
42	36, 40, 41	mp2an 699	. . . . . 6 |- ((F |` (A[,]B)) e. ((A[,]B)-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->RR))
43	33, 42	ax-mp 7	. . . . 5 |- (F |` (A[,]B)) e. ((A[,]B)-cn->RR)
44		isupivth.9	. . . . . 6 |- S = {x e. (A[,]B) | (F` x) = U}
45	37	eqeq1d 1486	. . . . . . 7 |- (x e. (A[,]B) -> (((F |` (A[,]B))` x) = U <-> (F` x) = U))
46	45	rabbii 1808	. . . . . 6 |- {x e. (A[,]B) | ((F |` (A[,]B))` x) = U} = {x e. (A[,]B) | (F` x) = U}
47	44, 46	eqtr4 1501	. . . . 5 |- S = {x e. (A[,]B) | ((F |` (A[,]B))` x) = U}
48	2, 3, 4, 5, 19, 25, 26, 43	ivthlem8 7288	. . . . . 6 |- (sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) e. (A(,)B) /\ ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )) = U)
49	48	pm3.27i 324	. . . . 5 |- ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )) = U
50	2, 3, 4, 5, 19, 25, 26, 43, 47, 49	ivthlem9 7289	. . . 4 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup(S, RR, < )
51	1, 50	eqtr4 1501	. . 3 |- C = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )
52	48	pm3.26i 320	. . 3 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) e. (A(,)B)
53	51, 52	eqeltr 1547	. 2 |- C e. (A(,)B)
54	51	fveq2i 3733	. . 3 |- ((F |` (A[,]B))` C) = ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ))
55		ioossicc 6398	. . . . 5 |- (A(,)B) (_ (A[,]B)
56	55, 53	sselii 2069	. . . 4 |- C e. (A[,]B)
57		fvres 3740	. . . 4 |- (C e. (A[,]B) -> ((F |` (A[,]B))` C) = (F` C))
58	56, 57	ax-mp 7	. . 3 |- ((F |` (A[,]B))` C) = (F` C)
59	54, 58, 49	3eqtr3 1506	. 2 |- (F` C) = U
60	53, 59	pm3.2i 285	1 |- (C e. (A(,)B) /\ (F` C) = U)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  {crab 1651   (_ wss 2050   class class class wbr 2624   |` cres 3178  ` cfv 3188  (class class class)co 3969  supcsup 4582  CCcc 5244  RRcr 5245   <_ cle 5307   < clt 5498  (,)cioo 6358  [,]cicc 6361  -cn->ccncf 7262

This theorem is referenced by:  dsupivthlem 7291  reeff1olem1 7424

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-q 6257  df-rp 6282  df-seq1 6309  df-ioo 6362  df-icc 6365  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-cncf 7263

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