Theorem supxr 7290

Description: The supremum of a set of extended reals.

Assertion

Ref	Expression
supxr	|- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem supxr

Step	Hyp	Ref	Expression
1		xrub 7289	. . . . 5 |- ((A C_ RR* /\ B e. RR*) -> (A.x e. RR (x < B -> E.y e. A x < y) <-> A.x e. RR* (x < B -> E.y e. A x < y)))
2	1	biimpd 170	. . . 4 |- ((A C_ RR* /\ B e. RR*) -> (A.x e. RR (x < B -> E.y e. A x < y) -> A.x e. RR* (x < B -> E.y e. A x < y)))
3	2	anim2d 620	. . 3 |- ((A C_ RR* /\ B e. RR*) -> ((A.x e. A -. B < x /\ A.x e. RR (x < B -> E.y e. A x < y)) -> (A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y))))
4	3	imp 377	. 2 |- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR (x < B -> E.y e. A x < y))) -> (A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y)))
5		xrsupss 7287	. . . . . . 7 |- (A C_ RR* -> E.z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y)))
6	5	anim2i 362	. . . . . 6 |- ((B e. RR* /\ A C_ RR*) -> (B e. RR* /\ E.z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))))
7	6	ancoms 484	. . . . 5 |- ((A C_ RR* /\ B e. RR*) -> (B e. RR* /\ E.z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))))
8		xrltso 6729	. . . . . . 7 |- < Or RR*
9	8	supeu 5668	. . . . . 6 |- (E.z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y)) -> E!z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y)))
10	9	anim2i 362	. . . . 5 |- ((B e. RR* /\ E.z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))) -> (B e. RR* /\ E!z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))))
11		breq1 3341	. . . . . . . . 9 |- (z = B -> (z < x <-> B < x))
12	11	notbid 673	. . . . . . . 8 |- (z = B -> (-. z < x <-> -. B < x))
13	12	ralbidv 2123	. . . . . . 7 |- (z = B -> (A.x e. A -. z < x <-> A.x e. A -. B < x))
14		breq2 3342	. . . . . . . . 9 |- (z = B -> (x < z <-> x < B))
15	14	imbi1d 675	. . . . . . . 8 |- (z = B -> ((x < z -> E.y e. A x < y) <-> (x < B -> E.y e. A x < y)))
16	15	ralbidv 2123	. . . . . . 7 |- (z = B -> (A.x e. RR* (x < z -> E.y e. A x < y) <-> A.x e. RR* (x < B -> E.y e. A x < y)))
17	13, 16	anbi12d 690	. . . . . 6 |- (z = B -> ((A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y)) <-> (A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y))))
18	17	reuuni2 3811	. . . . 5 |- ((B e. RR* /\ E!z e. RR* (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))) -> ((A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y)) <-> U.{z e. RR* | (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))} = B))
19	7, 10, 18	3syl 24	. . . 4 |- ((A C_ RR* /\ B e. RR*) -> ((A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y)) <-> U.{z e. RR* | (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))} = B))
20	19	biimpa 460	. . 3 |- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y))) -> U.{z e. RR* | (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))} = B)
21		df-sup 5664	. . 3 |- sup(A, RR*, < ) = U.{z e. RR* | (A.x e. A -. z < x /\ A.x e. RR* (x < z -> E.y e. A x < y))}
22	20, 21	syl5eq 1940	. 2 |- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR* (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)
23	4, 22	syldan 516	1 |- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108   C_ wss 2593  U.cuni 3177   class class class wbr 3338  supcsup 5663  RRcr 6385  RR*cxr 6652   < clt 6653

This theorem is referenced by:  supxr2 7291  supxrun 7294  supxrpnf 7297  supxrunb1 7298  supxrunb2 7299  xrsup0 7306

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

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