Theorem isupivth 7233

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 5562	. . . . . . . . . 10 |- A <_ B
8		lbicc2t 6345	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
9	2, 3, 7, 8	mp3an 914	. . . . . . . . 9 |- A e. (A[,]B)
10		fvres 3725	. . . . . . . . 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 2621	. . . . . . 7 |- (((F |` (A[,]B))` A) < U <-> (F` A) < U)
13		ubicc2t 6346	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
14	2, 3, 7, 13	mp3an 914	. . . . . . . . 9 |- B e. (A[,]B)
15		fvres 3725	. . . . . . . . 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 2622	. . . . . . 7 |- (U < ((F |` (A[,]B))` B) <-> U < (F` B))
18	12, 17	anbi12i 482	. . . . . 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 3725	. . . . . . . . 9 |- (c e. (A[,]B) -> ((F |` (A[,]B))` c) = (F` c))
21	20	breq1d 2624	. . . . . . . 8 |- (c e. (A[,]B) -> (((F |` (A[,]B))` c) <_ U <-> (F` c) <_ U))
22	21	rabbii 1801	. . . . . . 7 |- {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | (F` c) <_ U}
23		supeq1 4555	. . . . . . 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 1476	. . . . 5 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < )
26		eqid 1473	. . . . 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 2076	. . . . . . . 8 |- CC (_ CC
30		isupivth.5	. . . . . . . 8 |- (A[,]B) (_ D
31		rescncf 7215	. . . . . . . 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 914	. . . . . . 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 2069	. . . . . . . 8 |- (A[,]B) (_ CC
35		axresscn 5248	. . . . . . . 8 |- RR (_ CC
36	34, 29, 35	3pm3.2i 817	. . . . . . 7 |- ((A[,]B) (_ CC /\ CC (_ CC /\ RR (_ CC)
37		fvres 3725	. . . . . . . . 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 1545	. . . . . . . 8 |- (x e. (A[,]B) -> ((F |` (A[,]B))` x) e. RR)
40	39	rgen 1695	. . . . . . 7 |- A.x e. (A[,]B)((F |` (A[,]B))` x) e. RR
41		cncffvrn 7216	. . . . . . 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 696	. . . . . 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 1480	. . . . . . 7 |- (x e. (A[,]B) -> (((F |` (A[,]B))` x) = U <-> (F` x) = U))
46	45	rabbii 1801	. . . . . 6 |- {x e. (A[,]B) | ((F |` (A[,]B))` x) = U} = {x e. (A[,]B) | (F` x) = U}
47	44, 46	eqtr4 1495	. . . . 5 |- S = {x e. (A[,]B) | ((F |` (A[,]B))` x) = U}
48	2, 3, 4, 5, 19, 25, 26, 43	ivthlem8 7231	. . . . . 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 7232	. . . 4 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup(S, RR, < )
51	1, 50	eqtr4 1495	. . 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 1541	. 2 |- C e. (A(,)B)
54	51	fveq2i 3718	. . 3 |- ((F |` (A[,]B))` C) = ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ))
55		ioossicc 6338	. . . . 5 |- (A(,)B) (_ (A[,]B)
56	55, 53	sselii 2062	. . . 4 |- C e. (A[,]B)
57		fvres 3725	. . . 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 1500	. 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 774   = wceq 954   e. wcel 956  A.wral 1642  {crab 1645   (_ wss 2043   class class class wbr 2614   |` cres 3167  ` cfv 3177  (class class class)co 3954  supcsup 4553  CCcc 5212  RRcr 5213   <_ cle 5275   < clt 5466  (,)cioo 6302  [,]cicc 6305  -cn->ccncf 7205

This theorem is referenced by:  dsupivthlem 7234  reeff1olem1 7372

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-n0 6055  df-z 6091  df-q 6202  df-rp 6227  df-seq1 6253  df-ioo 6306  df-icc 6309  df-exp 6509  df-sqr 6608  df-re 6690  df-im 6691  df-cj 6692  df-abs 6693  df-cncf 7206

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