Theorem dsupivthlem 8553

Description: Lemma for dsupivthi 8554.

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}
dsupivth.10	|- ((F` B) < U /\ U < (F` A))
dsupivth.11	|- C = sup(S, RR, < )
dsupivthlem.12	|- G = {<.a, b>. | (a e. D /\ b = -u(F` a))}

Assertion

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

Distinct variable groups:   A,a,b,x   B,a,b,x   C,a,b   D,a,b,x   F,a,b,x   x,G   x,U

Proof of Theorem dsupivthlem

Step	Hyp	Ref	Expression
1		isupivth.1	. . . 4 |- A e. RR
2		isupivth.2	. . . 4 |- B e. RR
3		isupivth.3	. . . . 5 |- U e. RR
4	3	renegcli 6576	. . . 4 |- -uU e. RR
5		isupivth.4	. . . 4 |- A < B
6		isupivth.5	. . . 4 |- (A[,]B) C_ D
7		isupivth.6	. . . 4 |- D C_ CC
8		isupivth.7	. . . . 5 |- F e. (D-cn->CC)
9		dsupivthlem.12	. . . . 5 |- G = {<.a, b>. | (a e. D /\ b = -u(F` a))}
10	7, 8, 9	negfcncfi 8531	. . . 4 |- G e. (D-cn->CC)
11	6	sseli 2617	. . . . . 6 |- (x e. (A[,]B) -> x e. D)
12		fveq2 4681	. . . . . . . 8 |- (a = x -> (F` a) = (F` x))
13	12	negeqd 6516	. . . . . . 7 |- (a = x -> -u(F` a) = -u(F` x))
14		negex 6522	. . . . . . 7 |- -u(F` x) e. _V
15	13, 9, 14	fvopab4 4743	. . . . . 6 |- (x e. D -> (G` x) = -u(F` x))
16	11, 15	syl 12	. . . . 5 |- (x e. (A[,]B) -> (G` x) = -u(F` x))
17		isupivth.8	. . . . . 6 |- (x e. (A[,]B) -> (F` x) e. RR)
18		renegcl 6600	. . . . . 6 |- ((F` x) e. RR -> -u(F` x) e. RR)
19	17, 18	syl 12	. . . . 5 |- (x e. (A[,]B) -> -u(F` x) e. RR)
20	16, 19	eqeltrd 1971	. . . 4 |- (x e. (A[,]B) -> (G` x) e. RR)
21		isupivth.9	. . . . 5 |- S = {x e. (A[,]B) | (F` x) = U}
22	16	eqeq1d 1892	. . . . . . 7 |- (x e. (A[,]B) -> ((G` x) = -uU <-> -u(F` x) = -uU))
23		neg11 6569	. . . . . . . 8 |- (((F` x) e. CC /\ U e. CC) -> (-u(F` x) = -uU <-> (F` x) = U))
24	17	recnd 6468	. . . . . . . 8 |- (x e. (A[,]B) -> (F` x) e. CC)
25	3	recni 6467	. . . . . . . 8 |- U e. CC
26	23, 24, 25	sylancl 525	. . . . . . 7 |- (x e. (A[,]B) -> (-u(F` x) = -uU <-> (F` x) = U))
27	22, 26	bitr2d 588	. . . . . 6 |- (x e. (A[,]B) -> ((F` x) = U <-> (G` x) = -uU))
28	27	rabbiia 2285	. . . . 5 |- {x e. (A[,]B) | (F` x) = U} = {x e. (A[,]B) | (G` x) = -uU}
29	21, 28	eqtri 1908	. . . 4 |- S = {x e. (A[,]B) | (G` x) = -uU}
30	1, 2, 5	ltleii 6756	. . . . . . . . 9 |- A <_ B
31		lbicc2 7573	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
32	1, 2, 30, 31	mp3an 1191	. . . . . . . 8 |- A e. (A[,]B)
33	6, 32	sselii 2618	. . . . . . 7 |- A e. D
34		fveq2 4681	. . . . . . . . 9 |- (a = A -> (F` a) = (F` A))
35	34	negeqd 6516	. . . . . . . 8 |- (a = A -> -u(F` a) = -u(F` A))
36		negex 6522	. . . . . . . 8 |- -u(F` A) e. _V
37	35, 9, 36	fvopab4 4743	. . . . . . 7 |- (A e. D -> (G` A) = -u(F` A))
38	33, 37	ax-mp 7	. . . . . 6 |- (G` A) = -u(F` A)
39		dsupivth.10	. . . . . . . 8 |- ((F` B) < U /\ U < (F` A))
40	39	simpri 351	. . . . . . 7 |- U < (F` A)
41	17	rgen 2159	. . . . . . . . 9 |- A.x e. (A[,]B)(F` x) e. RR
42		fveq2 4681	. . . . . . . . . . 11 |- (x = A -> (F` x) = (F` A))
43	42	eleq1d 1963	. . . . . . . . . 10 |- (x = A -> ((F` x) e. RR <-> (F` A) e. RR))
44	43	rcla4v 2376	. . . . . . . . 9 |- (A e. (A[,]B) -> (A.x e. (A[,]B)(F` x) e. RR -> (F` A) e. RR))
45	32, 41, 44	mp2 54	. . . . . . . 8 |- (F` A) e. RR
46	3, 45	ltnegi 6783	. . . . . . 7 |- (U < (F` A) <-> -u(F` A) < -uU)
47	40, 46	mpbi 206	. . . . . 6 |- -u(F` A) < -uU
48	38, 47	eqbrtri 3356	. . . . 5 |- (G` A) < -uU
49	39	simpli 347	. . . . . . 7 |- (F` B) < U
50		ubicc2 7574	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
51	1, 2, 30, 50	mp3an 1191	. . . . . . . . 9 |- B e. (A[,]B)
52		fveq2 4681	. . . . . . . . . . 11 |- (x = B -> (F` x) = (F` B))
53	52	eleq1d 1963	. . . . . . . . . 10 |- (x = B -> ((F` x) e. RR <-> (F` B) e. RR))
54	53	rcla4v 2376	. . . . . . . . 9 |- (B e. (A[,]B) -> (A.x e. (A[,]B)(F` x) e. RR -> (F` B) e. RR))
55	51, 41, 54	mp2 54	. . . . . . . 8 |- (F` B) e. RR
56	55, 3	ltnegi 6783	. . . . . . 7 |- ((F` B) < U <-> -uU < -u(F` B))
57	49, 56	mpbi 206	. . . . . 6 |- -uU < -u(F` B)
58	6, 51	sselii 2618	. . . . . . 7 |- B e. D
59		fveq2 4681	. . . . . . . . 9 |- (a = B -> (F` a) = (F` B))
60	59	negeqd 6516	. . . . . . . 8 |- (a = B -> -u(F` a) = -u(F` B))
61		negex 6522	. . . . . . . 8 |- -u(F` B) e. _V
62	60, 9, 61	fvopab4 4743	. . . . . . 7 |- (B e. D -> (G` B) = -u(F` B))
63	58, 62	ax-mp 7	. . . . . 6 |- (G` B) = -u(F` B)
64	57, 63	breqtrri 3362	. . . . 5 |- -uU < (G` B)
65	48, 64	pm3.2i 307	. . . 4 |- ((G` A) < -uU /\ -uU < (G` B))
66		dsupivth.11	. . . 4 |- C = sup(S, RR, < )
67	1, 2, 4, 5, 6, 7, 10, 20, 29, 65, 66	isupivthi 8552	. . 3 |- (C e. (A(,)B) /\ (G` C) = -uU)
68	67	simpli 347	. 2 |- C e. (A(,)B)
69		ioossicc 7566	. . . . . 6 |- (A(,)B) C_ (A[,]B)
70	69, 68	sselii 2618	. . . . 5 |- C e. (A[,]B)
71	6	sseli 2617	. . . . . 6 |- (C e. (A[,]B) -> C e. D)
72		fveq2 4681	. . . . . . . 8 |- (a = C -> (F` a) = (F` C))
73	72	negeqd 6516	. . . . . . 7 |- (a = C -> -u(F` a) = -u(F` C))
74		negex 6522	. . . . . . 7 |- -u(F` C) e. _V
75	73, 9, 74	fvopab4 4743	. . . . . 6 |- (C e. D -> (G` C) = -u(F` C))
76	71, 75	syl 12	. . . . 5 |- (C e. (A[,]B) -> (G` C) = -u(F` C))
77	70, 76	ax-mp 7	. . . 4 |- (G` C) = -u(F` C)
78	67	simpri 351	. . . 4 |- (G` C) = -uU
79	77, 78	eqtr3i 1910	. . 3 |- -u(F` C) = -uU
80		ssid 2634	. . . . . 6 |- CC C_ CC
81		cncffvelrn 8530	. . . . . 6 |- ((D C_ CC /\ CC C_ CC /\ F e. (D-cn->CC)) -> (C e. D -> (F` C) e. CC))
82	7, 80, 8, 81	mp3an 1191	. . . . 5 |- (C e. D -> (F` C) e. CC)
83		neg11 6569	. . . . . 6 |- (((F` C) e. CC /\ U e. CC) -> (-u(F` C) = -uU <-> (F` C) = U))
84	25, 83	mpan2 760	. . . . 5 |- ((F` C) e. CC -> (-u(F` C) = -uU <-> (F` C) = U))
85	71, 82, 84	3syl 24	. . . 4 |- (C e. (A[,]B) -> (-u(F` C) = -uU <-> (F` C) = U))
86	70, 85	ax-mp 7	. . 3 |- (-u(F` C) = -uU <-> (F` C) = U)
87	79, 86	mpbi 206	. 2 |- (F` C) = U
88	68, 87	pm3.2i 307	1 |- (C e. (A(,)B) /\ (F` C) = U)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385  -ucneg 6446   <_ cle 6448   < clt 6653  (,)cioo 7524  [,]cicc 7527  -cn->ccncf 8524

This theorem is referenced by:  dsupivthi 8554

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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