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Unicode version
Theorem ioo2bl 9190
Description: An open interval of reals in terms of a ball.
Hypothesis
Ref Expression
remet.1 |- D = ((abs o. - ) |` (RR X. RR))
Assertion
Ref Expression
ioo2bl |- ((A e. RR /\ B e. RR /\ A < B) -> (A(,)B) = (((A + B) / 2)( ball ` D)((B - A) / 2)))
Proof of Theorem ioo2bl
StepHypRef Expression
1 readdcl 6455 . . . . . . 7 |- ((B e. RR /\ A e. RR) -> (B + A) e. RR)
2 rehalfcl 7220 . . . . . . 7 |- ((B + A) e. RR -> ((B + A) / 2) e. RR)
31, 2syl 12 . . . . . 6 |- ((B e. RR /\ A e. RR) -> ((B + A) / 2) e. RR)
433adant3 896 . . . . 5 |- ((B e. RR /\ A e. RR /\ A < B) -> ((B + A) / 2) e. RR)
5 resubcl 6601 . . . . . . 7 |- ((B e. RR /\ A e. RR) -> (B - A) e. RR)
6 rehalfcl 7220 . . . . . . 7 |- ((B - A) e. RR -> ((B - A) / 2) e. RR)
75, 6syl 12 . . . . . 6 |- ((B e. RR /\ A e. RR) -> ((B - A) / 2) e. RR)
873adant3 896 . . . . 5 |- ((B e. RR /\ A e. RR /\ A < B) -> ((B - A) / 2) e. RR)
9 posdif 6843 . . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A < B <-> 0 < (B - A)))
109ancoms 484 . . . . . . 7 |- ((B e. RR /\ A e. RR) -> (A < B <-> 0 < (B - A)))
11 halfpos2 7223 . . . . . . . 8 |- ((B - A) e. RR -> (0 < (B - A) <-> 0 < ((B - A) / 2)))
125, 11syl 12 . . . . . . 7 |- ((B e. RR /\ A e. RR) -> (0 < (B - A) <-> 0 < ((B - A) / 2)))
1310, 12bitrd 587 . . . . . 6 |- ((B e. RR /\ A e. RR) -> (A < B <-> 0 < ((B - A) / 2)))
1413biimp3a 1194 . . . . 5 |- ((B e. RR /\ A e. RR /\ A < B) -> 0 < ((B - A) / 2))
15 remet.1 . . . . . 6 |- D = ((abs o. - ) |` (RR X. RR))
1615bl2ioo 9189 . . . . 5 |- ((((B + A) / 2) e. RR /\ ((B - A) / 2) e. RR /\ 0 < ((B - A) / 2)) -> (((B + A) / 2)( ball ` D)((B - A) / 2)) = ((((B + A) / 2) - ((B - A) / 2))(,)(((B + A) / 2) + ((B - A) / 2))))
174, 8, 14, 16syl111anc 1100 . . . 4 |- ((B e. RR /\ A e. RR /\ A < B) -> (((B + A) / 2)( ball ` D)((B - A) / 2)) = ((((B + A) / 2) - ((B - A) / 2))(,)(((B + A) / 2) + ((B - A) / 2))))
18 pnncan 6647 . . . . . . . . . . 11 |- ((B e. CC /\ A e. CC /\ A e. CC) -> ((B + A) - (B - A)) = (A + A))
19183anidm23 1156 . . . . . . . . . 10 |- ((B e. CC /\ A e. CC) -> ((B + A) - (B - A)) = (A + A))
20 2times 7188 . . . . . . . . . . 11 |- (A e. CC -> (2 x. A) = (A + A))
2120adantl 424 . . . . . . . . . 10 |- ((B e. CC /\ A e. CC) -> (2 x. A) = (A + A))
2219, 21eqtr4d 1928 . . . . . . . . 9 |- ((B e. CC /\ A e. CC) -> ((B + A) - (B - A)) = (2 x. A))
2322opreq1d 4897 . . . . . . . 8 |- ((B e. CC /\ A e. CC) -> (((B + A) - (B - A)) / 2) = ((2 x. A) / 2))
24 addcl 6454 . . . . . . . . 9 |- ((B e. CC /\ A e. CC) -> (B + A) e. CC)
25 subcl 6524 . . . . . . . . 9 |- ((B e. CC /\ A e. CC) -> (B - A) e. CC)
26 2cn 7164 . . . . . . . . . . 11 |- 2 e. CC
27 2ne0 7174 . . . . . . . . . . 11 |- 2 =/= 0
2826, 27pm3.2i 307 . . . . . . . . . 10 |- (2 e. CC /\ 2 =/= 0)
29 divsubdir 6951 . . . . . . . . . 10 |- (((B + A) e. CC /\ (B - A) e. CC /\ (2 e. CC /\ 2 =/= 0)) -> (((B + A) - (B - A)) / 2) = (((B + A) / 2) - ((B - A) / 2)))
3028, 29mp3an3 1180 . . . . . . . . 9 |- (((B + A) e. CC /\ (B - A) e. CC) -> (((B + A) - (B - A)) / 2) = (((B + A) / 2) - ((B - A) / 2)))
3124, 25, 30syl11anc 524 . . . . . . . 8 |- ((B e. CC /\ A e. CC) -> (((B + A) - (B - A)) / 2) = (((B + A) / 2) - ((B - A) / 2)))
32 divcan3 6938 . . . . . . . . . 10 |- ((A e. CC /\ 2 e. CC /\ 2 =/= 0) -> ((2 x. A) / 2) = A)
3326, 27, 32mp3an23 1183 . . . . . . . . 9 |- (A e. CC -> ((2 x. A) / 2) = A)
3433adantl 424 . . . . . . . 8 |- ((B e. CC /\ A e. CC) -> ((2 x. A) / 2) = A)
3523, 31, 343eqtr3d 1934 . . . . . . 7 |- ((B e. CC /\ A e. CC) -> (((B + A) / 2) - ((B - A) / 2)) = A)
36 ppncan 6648 . . . . . . . . . . 11 |- ((B e. CC /\ A e. CC /\ B e. CC) -> ((B + A) + (B - A)) = (B + B))
37363anidm13 1155 . . . . . . . . . 10 |- ((B e. CC /\ A e. CC) -> ((B + A) + (B - A)) = (B + B))
38 2times 7188 . . . . . . . . . . 11 |- (B e. CC -> (2 x. B) = (B + B))
3938adantr 425 . . . . . . . . . 10 |- ((B e. CC /\ A e. CC) -> (2 x. B) = (B + B))
4037, 39eqtr4d 1928 . . . . . . . . 9 |- ((B e. CC /\ A e. CC) -> ((B + A) + (B - A)) = (2 x. B))
4140opreq1d 4897 . . . . . . . 8 |- ((B e. CC /\ A e. CC) -> (((B + A) + (B - A)) / 2) = ((2 x. B) / 2))
42 divdir 6933 . . . . . . . . . 10 |- (((B + A) e. CC /\ (B - A) e. CC /\ (2 e. CC /\ 2 =/= 0)) -> (((B + A) + (B - A)) / 2) = (((B + A) / 2) + ((B - A) / 2)))
4328, 42mp3an3 1180 . . . . . . . . 9 |- (((B + A) e. CC /\ (B - A) e. CC) -> (((B + A) + (B - A)) / 2) = (((B + A) / 2) + ((B - A) / 2)))
4424, 25, 43syl11anc 524 . . . . . . . 8 |- ((B e. CC /\ A e. CC) -> (((B + A) + (B - A)) / 2) = (((B + A) / 2) + ((B - A) / 2)))
45 divcan3 6938 . . . . . . . . . 10 |- ((B e. CC /\ 2 e. CC /\ 2 =/= 0) -> ((2 x. B) / 2) = B)
4626, 27, 45mp3an23 1183 . . . . . . . . 9 |- (B e. CC -> ((2 x. B) / 2) = B)
4746adantr 425 . . . . . . . 8 |- ((B e. CC /\ A e. CC) -> ((2 x. B) / 2) = B)
4841, 44, 473eqtr3d 1934 . . . . . . 7 |- ((B e. CC /\ A e. CC) -> (((B + A) / 2) + ((B - A) / 2)) = B)
4935, 48opreq12d 4900 . . . . . 6 |- ((B e. CC /\ A e. CC) -> ((((B + A) / 2) - ((B - A) / 2))(,)(((B + A) / 2) + ((B - A) / 2))) = (A(,)B))
50 recn 6466 . . . . . 6 |- (B e. RR -> B e. CC)
51 recn 6466 . . . . . 6 |- (A e. RR -> A e. CC)
5249, 50, 51syl2an 503 . . . . 5 |- ((B e. RR /\ A e. RR) -> ((((B + A) / 2) - ((B - A) / 2))(,)(((B + A) / 2) + ((B - A) / 2))) = (A(,)B))
53523adant3 896 . . . 4 |- ((B e. RR /\ A e. RR /\ A < B) -> ((((B + A) / 2) - ((B - A) / 2))(,)(((B + A) / 2) + ((B - A) / 2))) = (A(,)B))
5417, 53eqtr2d 1926 . . 3 |- ((B e. RR /\ A e. RR /\ A < B) -> (A(,)B) = (((B + A) / 2)( ball ` D)((B - A) / 2)))
55543com12 1071 . 2 |- ((A e. RR /\ B e. RR /\ A < B) -> (A(,)B) = (((B + A) / 2)( ball ` D)((B - A) / 2)))
56 addcom 6458 . . . . . 6 |- ((A e. CC /\ B e. CC) -> (A + B) = (B + A))
5756, 51, 50syl2an 503 . . . . 5 |- ((A e. RR /\ B e. RR) -> (A + B) = (B + A))
5857opreq1d 4897 . . . 4 |- ((A e. RR /\ B e. RR) -> ((A + B) / 2) = ((B + A) / 2))
59583adant3 896 . . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> ((A + B) / 2) = ((B + A) / 2))
6059opreq1d 4897 . 2 |- ((A e. RR /\ B e. RR /\ A < B) -> (((A + B) / 2)( ball ` D)((B - A) / 2)) = (((B + A) / 2)( ball ` D)((B - A) / 2)))
6155, 60eqtr4d 1928 1 |- ((A e. RR /\ B e. RR /\ A < B) -> (A(,)B) = (((A + B) / 2)( ball ` D)((B - A) / 2)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338   X. cxp 3984   |` cres 3988   o. ccom 3990  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   < clt 6653  2c2 7145  (,)cioo 7524  abscabs 8000   ball cbl 9068
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-n0 7309  df-z 7345  df-ioo 7528  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-met 9070  df-bl 9072
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