Theorem ivthlem9 8551

Description: Lemma for isupivthi 8552.

Hypotheses

Ref	Expression
ivthlem4.1	|- A e. RR
ivthlem4.2	|- B e. RR
ivthlem4.3	|- U e. RR
ivthlem4.4	|- A < B
ivthlem4.5	|- ((F` A) < U /\ U < (F` B))
ivthlem4.6	|- C = sup(S, RR, < )
ivthlem4.7	|- S = {c e. (A[,]B) | (F` c) <_ U}
ivthlem4.8	|- F e. ((A[,]B)-cn->RR)
ivthlem9.9	|- T = {d e. (A[,]B) | (F` d) = U}
ivthlem9.10	|- (F` C) = U

Assertion

Ref	Expression
ivthlem9	|- C = sup(T, RR, < )

Distinct variable groups:   A,c,d   B,c,d   C,d   F,c,d   U,c,d

Proof of Theorem ivthlem9

Step	Hyp	Ref	Expression
1		ivthlem4.1	. . . . 5 |- A e. RR
2		ivthlem4.2	. . . . 5 |- B e. RR
3		iccssre 7565	. . . . 5 |- ((A e. RR /\ B e. RR) -> (A[,]B) C_ RR)
4	1, 2, 3	mp2an 761	. . . 4 |- (A[,]B) C_ RR
5		ivthlem4.3	. . . . 5 |- U e. RR
6		ivthlem4.4	. . . . 5 |- A < B
7		ivthlem4.5	. . . . 5 |- ((F` A) < U /\ U < (F` B))
8		ivthlem4.6	. . . . 5 |- C = sup(S, RR, < )
9		ivthlem4.7	. . . . 5 |- S = {c e. (A[,]B) | (F` c) <_ U}
10		ivthlem4.8	. . . . 5 |- F e. ((A[,]B)-cn->RR)
11	1, 2, 5, 6, 7, 8, 9, 10	ivthlem5 8547	. . . 4 |- C e. (A[,]B)
12	4, 11	sselii 2618	. . 3 |- C e. RR
13		ivthlem9.9	. . . . . . 7 |- T = {d e. (A[,]B) | (F` d) = U}
14		ssrab2 2692	. . . . . . 7 |- {d e. (A[,]B) | (F` d) = U} C_ (A[,]B)
15	13, 14	eqsstri 2647	. . . . . 6 |- T C_ (A[,]B)
16		fveq2 4681	. . . . . . . . 9 |- (d = C -> (F` d) = (F` C))
17	16	eqeq1d 1892	. . . . . . . 8 |- (d = C -> ((F` d) = U <-> (F` C) = U))
18	17, 13	elrab2 2416	. . . . . . 7 |- (C e. T <-> (C e. (A[,]B) /\ (F` C) = U))
19		ivthlem9.10	. . . . . . 7 |- (F` C) = U
20	18, 11, 19	mpbir2an 800	. . . . . 6 |- C e. T
21		iccsupr 7567	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ T C_ (A[,]B) /\ C e. T) -> (T C_ RR /\ T =/= (/) /\ E.v e. RR A.w e. T w <_ v))
22	15, 20, 21	mp3an23 1183	. . . . 5 |- ((A e. RR /\ B e. RR) -> (T C_ RR /\ T =/= (/) /\ E.v e. RR A.w e. T w <_ v))
23	1, 2, 22	mp2an 761	. . . 4 |- (T C_ RR /\ T =/= (/) /\ E.v e. RR A.w e. T w <_ v)
24	23	suprclii 7270	. . 3 |- sup(T, RR, < ) e. RR
25	12, 24	letri3i 6749	. 2 |- (C = sup(T, RR, < ) <-> (C <_ sup(T, RR, < ) /\ sup(T, RR, < ) <_ C))
26	23	suprubii 7271	. . 3 |- (C e. T -> C <_ sup(T, RR, < ))
27	20, 26	ax-mp 7	. 2 |- C <_ sup(T, RR, < )
28		fveq2 4681	. . . . . . 7 |- (d = x -> (F` d) = (F` x))
29	28	eqeq1d 1892	. . . . . 6 |- (d = x -> ((F` d) = U <-> (F` x) = U))
30	29, 13	elrab2 2416	. . . . 5 |- (x e. T <-> (x e. (A[,]B) /\ (F` x) = U))
31		axresscn 6420	. . . . . . . . . 10 |- RR C_ CC
32	4, 31	sstri 2626	. . . . . . . . 9 |- (A[,]B) C_ CC
33		cncffvelrn 8530	. . . . . . . . 9 |- (((A[,]B) C_ CC /\ RR C_ CC /\ F e. ((A[,]B)-cn->RR)) -> (x e. (A[,]B) -> (F` x) e. RR))
34	32, 31, 10, 33	mp3an 1191	. . . . . . . 8 |- (x e. (A[,]B) -> (F` x) e. RR)
35		eqle 6746	. . . . . . . . 9 |- (((F` x) e. RR /\ (F` x) = U) -> (F` x) <_ U)
36	35	ex 402	. . . . . . . 8 |- ((F` x) e. RR -> ((F` x) = U -> (F` x) <_ U))
37	34, 36	syl 12	. . . . . . 7 |- (x e. (A[,]B) -> ((F` x) = U -> (F` x) <_ U))
38		fveq2 4681	. . . . . . . . . . 11 |- (c = x -> (F` c) = (F` x))
39	38	breq1d 3348	. . . . . . . . . 10 |- (c = x -> ((F` c) <_ U <-> (F` x) <_ U))
40	39, 9	elrab2 2416	. . . . . . . . 9 |- (x e. S <-> (x e. (A[,]B) /\ (F` x) <_ U))
41	1, 2, 5, 6, 7, 8, 9, 10	ivthlem4 8546	. . . . . . . . . . . 12 |- ((S C_ RR /\ S =/= (/) /\ E.v e. RR A.w e. S w <_ v) /\ A e. S)
42	41	simpli 347	. . . . . . . . . . 11 |- (S C_ RR /\ S =/= (/) /\ E.v e. RR A.w e. S w <_ v)
43	42	suprubii 7271	. . . . . . . . . 10 |- (x e. S -> x <_ sup(S, RR, < ))
44	43, 8	syl6breqr 3377	. . . . . . . . 9 |- (x e. S -> x <_ C)
45	40, 44	sylbir 218	. . . . . . . 8 |- ((x e. (A[,]B) /\ (F` x) <_ U) -> x <_ C)
46	45	ex 402	. . . . . . 7 |- (x e. (A[,]B) -> ((F` x) <_ U -> x <_ C))
47	37, 46	syld 30	. . . . . 6 |- (x e. (A[,]B) -> ((F` x) = U -> x <_ C))
48	47	imp 377	. . . . 5 |- ((x e. (A[,]B) /\ (F` x) = U) -> x <_ C)
49	30, 48	sylbi 216	. . . 4 |- (x e. T -> x <_ C)
50	49	rgen 2159	. . 3 |- A.x e. T x <_ C
51	23	suprleubii 7274	. . 3 |- ((C e. RR /\ A.x e. T x <_ C) -> sup(T, RR, < ) <_ C)
52	12, 50, 51	mp2an 761	. 2 |- sup(T, RR, < ) <_ C
53	25, 27, 52	mpbir2an 800	1 |- C = sup(T, RR, < )

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   C_ wss 2593  (/)c0 2875   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385   <_ cle 6448   < clt 6653  [,]cicc 7527  -cn->ccncf 8524

This theorem is referenced by:  isupivthi 8552

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-icc 7531  df-cncf 8525

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