Theorem isupivthi 8552

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) C_ D
isupivth.6	|- D C_ 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
isupivthi	|- (C e. (A(,)B) /\ (F` C) = U)

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

Proof of Theorem isupivthi

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	ltleii 6756	. . . . . . . . . 10 |- A <_ B
8		lbicc2 7573	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
9	2, 3, 7, 8	mp3an 1191	. . . . . . . . 9 |- A e. (A[,]B)
10		fvres 4691	. . . . . . . . 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 3345	. . . . . . 7 |- (((F |` (A[,]B))` A) < U <-> (F` A) < U)
13		ubicc2 7574	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
14	2, 3, 7, 13	mp3an 1191	. . . . . . . . 9 |- B e. (A[,]B)
15		fvres 4691	. . . . . . . . 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 3346	. . . . . . 7 |- (U < ((F |` (A[,]B))` B) <-> U < (F` B))
18	12, 17	anbi12i 540	. . . . . 6 |- ((((F |` (A[,]B))` A) < U /\ U < ((F |` (A[,]B))` B)) <-> ((F` A) < U /\ U < (F` B)))
19	6, 18	mpbir 207	. . . . 5 |- (((F |` (A[,]B))` A) < U /\ U < ((F |` (A[,]B))` B))
20		fvres 4691	. . . . . . . . 9 |- (c e. (A[,]B) -> ((F |` (A[,]B))` c) = (F` c))
21	20	breq1d 3348	. . . . . . . 8 |- (c e. (A[,]B) -> (((F |` (A[,]B))` c) <_ U <-> (F` c) <_ U))
22	21	rabbiia 2285	. . . . . . 7 |- {c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U} = {c e. (A[,]B) | (F` c) <_ U}
23		supeq1 5665	. . . . . . 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 1888	. . . . 5 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup({c e. (A[,]B) | ((F |` (A[,]B))` c) <_ U}, RR, < )
26		eqid 1884	. . . . 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 C_ CC
29		ssid 2634	. . . . . . . 8 |- CC C_ CC
30		isupivth.5	. . . . . . . 8 |- (A[,]B) C_ D
31		rescncf 8534	. . . . . . . 8 |- ((D C_ CC /\ CC C_ CC /\ (A[,]B) C_ D) -> (F e. (D-cn->CC) -> (F |` (A[,]B)) e. ((A[,]B)-cn->CC)))
32	28, 29, 30, 31	mp3an 1191	. . . . . . 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 2626	. . . . . . . 8 |- (A[,]B) C_ CC
35		axresscn 6420	. . . . . . . 8 |- RR C_ CC
36	34, 29, 35	3pm3.2i 1048	. . . . . . 7 |- ((A[,]B) C_ CC /\ CC C_ CC /\ RR C_ CC)
37		fvres 4691	. . . . . . . . 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 1971	. . . . . . . 8 |- (x e. (A[,]B) -> ((F |` (A[,]B))` x) e. RR)
40	39	rgen 2159	. . . . . . 7 |- A.x e. (A[,]B)((F |` (A[,]B))` x) e. RR
41		cncffvrn 8535	. . . . . . 7 |- ((((A[,]B) C_ CC /\ CC C_ CC /\ RR C_ 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 761	. . . . . 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 1892	. . . . . . 7 |- (x e. (A[,]B) -> (((F |` (A[,]B))` x) = U <-> (F` x) = U))
46	45	rabbiia 2285	. . . . . 6 |- {x e. (A[,]B) | ((F |` (A[,]B))` x) = U} = {x e. (A[,]B) | (F` x) = U}
47	44, 46	eqtr4i 1911	. . . . 5 |- S = {x e. (A[,]B) | ((F |` (A[,]B))` x) = U}
48	2, 3, 4, 5, 19, 25, 26, 43	ivthlem8 8550	. . . . . 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	simpri 351	. . . . 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 8551	. . . 4 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) = sup(S, RR, < )
51	1, 50	eqtr4i 1911	. . 3 |- C = sup({c e. (A[,]B) | (F` c) <_ U}, RR, < )
52	48	simpli 347	. . 3 |- sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ) e. (A(,)B)
53	51, 52	eqeltri 1967	. 2 |- C e. (A(,)B)
54	51	fveq2i 4684	. . 3 |- ((F |` (A[,]B))` C) = ((F |` (A[,]B))` sup({c e. (A[,]B) | (F` c) <_ U}, RR, < ))
55		ioossicc 7566	. . . . 5 |- (A(,)B) C_ (A[,]B)
56	55, 53	sselii 2618	. . . 4 |- C e. (A[,]B)
57		fvres 4691	. . . 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	3eqtr3i 1918	. 2 |- (F` C) = U
60	53, 59	pm3.2i 307	1 |- (C e. (A(,)B) /\ (F` C) = U)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593   class class class wbr 3338   |` cres 3988  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385   <_ cle 6448   < clt 6653  (,)cioo 7524  [,]cicc 7527  -cn->ccncf 8524

This theorem is referenced by:  dsupivthlem 8553  reeff1olem1 8689

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-ioo 7528  df-icc 7531  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-cncf 8525

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