Theorem supxrun 7294

Description: The supremum of the union of two sets of extended reals equals the largest of their suprema.

Assertion

Ref	Expression
supxrun	|- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> sup((A u. B), RR*, < ) = sup(B, RR*, < ))

Proof of Theorem supxrun

Step	Hyp	Ref	Expression
1		unss 2780	. . . 4 |- ((A C_ RR* /\ B C_ RR*) <-> (A u. B) C_ RR*)
2	1	biimpi 168	. . 3 |- ((A C_ RR* /\ B C_ RR*) -> (A u. B) C_ RR*)
3	2	3adant3 896	. 2 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (A u. B) C_ RR*)
4		supxrcl 7293	. . 3 |- (B C_ RR* -> sup(B, RR*, < ) e. RR*)
5	4	3ad2ant2 898	. 2 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> sup(B, RR*, < ) e. RR*)
6		xrsupss 7287	. . . . . . . 8 |- (A C_ RR* -> E.y e. RR* (A.z e. A -. y < z /\ A.z e. RR* (z < y -> E.w e. A z < w)))
7		xrltso 6729	. . . . . . . . 9 |- < Or RR*
8	7	supub 5670	. . . . . . . 8 |- (E.y e. RR* (A.z e. A -. y < z /\ A.z e. RR* (z < y -> E.w e. A z < w)) -> (x e. A -> -. sup(A, RR*, < ) < x))
9	6, 8	syl 12	. . . . . . 7 |- (A C_ RR* -> (x e. A -> -. sup(A, RR*, < ) < x))
10	9	3ad2ant1 897	. . . . . 6 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (x e. A -> -. sup(A, RR*, < ) < x))
11		supxrcl 7293	. . . . . . . . . . . . 13 |- (A C_ RR* -> sup(A, RR*, < ) e. RR*)
12	11	ad2antrr 440	. . . . . . . . . . . 12 |- (((A C_ RR* /\ B C_ RR*) /\ x e. A) -> sup(A, RR*, < ) e. RR*)
13	4	ad2antlr 441	. . . . . . . . . . . 12 |- (((A C_ RR* /\ B C_ RR*) /\ x e. A) -> sup(B, RR*, < ) e. RR*)
14		ssel2 2616	. . . . . . . . . . . . 13 |- ((A C_ RR* /\ x e. A) -> x e. RR*)
15	14	adantlr 429	. . . . . . . . . . . 12 |- (((A C_ RR* /\ B C_ RR*) /\ x e. A) -> x e. RR*)
16		xrlelttr 6737	. . . . . . . . . . . 12 |- ((sup(A, RR*, < ) e. RR* /\ sup(B, RR*, < ) e. RR* /\ x e. RR*) -> ((sup(A, RR*, < ) <_ sup(B, RR*, < ) /\ sup(B, RR*, < ) < x) -> sup(A, RR*, < ) < x))
17	12, 13, 15, 16	syl111anc 1100	. . . . . . . . . . 11 |- (((A C_ RR* /\ B C_ RR*) /\ x e. A) -> ((sup(A, RR*, < ) <_ sup(B, RR*, < ) /\ sup(B, RR*, < ) < x) -> sup(A, RR*, < ) < x))
18	17	expdimp 406	. . . . . . . . . 10 |- ((((A C_ RR* /\ B C_ RR*) /\ x e. A) /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (sup(B, RR*, < ) < x -> sup(A, RR*, < ) < x))
19	18	con3d 111	. . . . . . . . 9 |- ((((A C_ RR* /\ B C_ RR*) /\ x e. A) /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (-. sup(A, RR*, < ) < x -> -. sup(B, RR*, < ) < x))
20	19	exp41 413	. . . . . . . 8 |- (A C_ RR* -> (B C_ RR* -> (x e. A -> (sup(A, RR*, < ) <_ sup(B, RR*, < ) -> (-. sup(A, RR*, < ) < x -> -. sup(B, RR*, < ) < x)))))
21	20	com34 40	. . . . . . 7 |- (A C_ RR* -> (B C_ RR* -> (sup(A, RR*, < ) <_ sup(B, RR*, < ) -> (x e. A -> (-. sup(A, RR*, < ) < x -> -. sup(B, RR*, < ) < x)))))
22	21	3imp 1061	. . . . . 6 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (x e. A -> (-. sup(A, RR*, < ) < x -> -. sup(B, RR*, < ) < x)))
23	10, 22	mpdd 57	. . . . 5 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (x e. A -> -. sup(B, RR*, < ) < x))
24		xrsupss 7287	. . . . . . 7 |- (B C_ RR* -> E.y e. RR* (A.z e. B -. y < z /\ A.z e. RR* (z < y -> E.w e. B z < w)))
25	7	supub 5670	. . . . . . 7 |- (E.y e. RR* (A.z e. B -. y < z /\ A.z e. RR* (z < y -> E.w e. B z < w)) -> (x e. B -> -. sup(B, RR*, < ) < x))
26	24, 25	syl 12	. . . . . 6 |- (B C_ RR* -> (x e. B -> -. sup(B, RR*, < ) < x))
27	26	3ad2ant2 898	. . . . 5 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (x e. B -> -. sup(B, RR*, < ) < x))
28	23, 27	jaod 469	. . . 4 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> ((x e. A \/ x e. B) -> -. sup(B, RR*, < ) < x))
29		elun 2741	. . . 4 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
30	28, 29	syl5ib 223	. . 3 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> (x e. (A u. B) -> -. sup(B, RR*, < ) < x))
31	30	r19.21aiv 2175	. 2 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> A.x e. (A u. B) -. sup(B, RR*, < ) < x)
32		xrsupss 7287	. . . . . . 7 |- (B C_ RR* -> E.x e. RR* (A.z e. B -. x < z /\ A.z e. RR* (z < x -> E.y e. B z < y)))
33	7	suplub 5671	. . . . . . . 8 |- (E.x e. RR* (A.z e. B -. x < z /\ A.z e. RR* (z < x -> E.y e. B z < y)) -> ((x e. RR* /\ x < sup(B, RR*, < )) -> E.y e. B x < y))
34		rexr 6668	. . . . . . . 8 |- (x e. RR -> x e. RR*)
35	33, 34	sylani 513	. . . . . . 7 |- (E.x e. RR* (A.z e. B -. x < z /\ A.z e. RR* (z < x -> E.y e. B z < y)) -> ((x e. RR /\ x < sup(B, RR*, < )) -> E.y e. B x < y))
36	32, 35	syl 12	. . . . . 6 |- (B C_ RR* -> ((x e. RR /\ x < sup(B, RR*, < )) -> E.y e. B x < y))
37		elun2 2772	. . . . . . . 8 |- (y e. B -> y e. (A u. B))
38	37	anim1i 361	. . . . . . 7 |- ((y e. B /\ x < y) -> (y e. (A u. B) /\ x < y))
39	38	reximi2 2197	. . . . . 6 |- (E.y e. B x < y -> E.y e. (A u. B)x < y)
40	36, 39	syl6 25	. . . . 5 |- (B C_ RR* -> ((x e. RR /\ x < sup(B, RR*, < )) -> E.y e. (A u. B)x < y))
41	40	exp3a 405	. . . 4 |- (B C_ RR* -> (x e. RR -> (x < sup(B, RR*, < ) -> E.y e. (A u. B)x < y)))
42	41	r19.21aiv 2175	. . 3 |- (B C_ RR* -> A.x e. RR (x < sup(B, RR*, < ) -> E.y e. (A u. B)x < y))
43	42	3ad2ant2 898	. 2 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> A.x e. RR (x < sup(B, RR*, < ) -> E.y e. (A u. B)x < y))
44		supxr 7290	. 2 |- ((((A u. B) C_ RR* /\ sup(B, RR*, < ) e. RR*) /\ (A.x e. (A u. B) -. sup(B, RR*, < ) < x /\ A.x e. RR (x < sup(B, RR*, < ) -> E.y e. (A u. B)x < y))) -> sup((A u. B), RR*, < ) = sup(B, RR*, < ))
45	3, 5, 31, 43, 44	syl22anc 1101	1 |- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> sup((A u. B), RR*, < ) = sup(B, RR*, < ))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   u. cun 2591   C_ wss 2593   class class class wbr 3338  supcsup 5663  RRcr 6385   <_ cle 6448  RR*cxr 6652   < clt 6653

This theorem is referenced by:  supxrmnf 7296

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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