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Theorem difioo 29544
Description: The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.)
Assertion
Ref Expression
difioo |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) )
Proof of Theorem difioo
StepHypRef Expression
1 incom 3805 . . . 4 |- ( ( A (,) B ) i^i ( B [,) C ) ) = ( ( B [,) C ) i^i ( A (,) B ) )
2 joiniooico 29536 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( ( ( A (,) B ) i^i ( B [,) C ) ) = (/) /\ ( ( A (,) B ) u. ( B [,) C ) ) = ( A (,) C ) ) )
32anassrs 680 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( ( A (,) B ) i^i ( B [,) C ) ) = (/) /\ ( ( A (,) B ) u. ( B [,) C ) ) = ( A (,) C ) ) )
43simpld 475 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) B ) i^i ( B [,) C ) ) = (/) )
51, 4syl5eqr 2670 . . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( B [,) C ) i^i ( A (,) B ) ) = (/) )
63simprd 479 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) B ) u. ( B [,) C ) ) = ( A (,) C ) )
7 uncom 3757 . . . . 5 |- ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) B ) u. ( B [,) C ) )
87a1i 11 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) B ) u. ( B [,) C ) ) )
9 simpll1 1100 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> A e. RR* )
10 simpl3 1066 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> C e. RR* )
1110adantr 481 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> C e. RR* )
12 xrleid 11983 . . . . . . 7 |- ( A e. RR* -> A <_ A )
139, 12syl 17 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> A <_ A )
14 simpr 477 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> B <_ C )
15 ioossioo 12265 . . . . . 6 |- ( ( ( A e. RR* /\ C e. RR* ) /\ ( A <_ A /\ B <_ C ) ) -> ( A (,) B ) C_ ( A (,) C ) )
169, 11, 13, 14, 15syl22anc 1327 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( A (,) B ) C_ ( A (,) C ) )
17 ssequn2 3786 . . . . 5 |- ( ( A (,) B ) C_ ( A (,) C ) <-> ( ( A (,) C ) u. ( A (,) B ) ) = ( A (,) C ) )
1816, 17sylib 208 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) C ) u. ( A (,) B ) ) = ( A (,) C ) )
196, 8, 183eqtr4d 2666 . . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) C ) u. ( A (,) B ) ) )
20 difeq 29355 . . 3 |- ( ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) <-> ( ( ( B [,) C ) i^i ( A (,) B ) ) = (/) /\ ( ( B [,) C ) u. ( A (,) B ) ) = ( ( A (,) C ) u. ( A (,) B ) ) ) )
215, 19, 20sylanbrc 698 . 2 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ B <_ C ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) )
22 simpll1 1100 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> A e. RR* )
23 simpl2 1065 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> B e. RR* )
2423adantr 481 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> B e. RR* )
2522, 12syl 17 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> A <_ A )
2610adantr 481 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> C e. RR* )
27 simpr 477 . . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> C < B )
28 xrltle 11982 . . . . . . 7 |- ( ( C e. RR* /\ B e. RR* ) -> ( C < B -> C <_ B ) )
2928imp 445 . . . . . 6 |- ( ( ( C e. RR* /\ B e. RR* ) /\ C < B ) -> C <_ B )
3026, 24, 27, 29syl21anc 1325 . . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> C <_ B )
31 ioossioo 12265 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ A /\ C <_ B ) ) -> ( A (,) C ) C_ ( A (,) B ) )
3222, 24, 25, 30, 31syl22anc 1327 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( A (,) C ) C_ ( A (,) B ) )
33 ssdif0 3942 . . . 4 |- ( ( A (,) C ) C_ ( A (,) B ) <-> ( ( A (,) C ) \ ( A (,) B ) ) = (/) )
3432, 33sylib 208 . . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = (/) )
35 ico0 12221 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( ( B [,) C ) = (/) <-> C <_ B ) )
3635biimpar 502 . . . 4 |- ( ( ( B e. RR* /\ C e. RR* ) /\ C <_ B ) -> ( B [,) C ) = (/) )
3724, 26, 30, 36syl21anc 1325 . . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( B [,) C ) = (/) )
3834, 37eqtr4d 2659 . 2 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) /\ C < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) )
39 xrlelttric 29517 . . 3 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C \/ C < B ) )
4023, 10, 39syl2anc 693 . 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> ( B <_ C \/ C < B ) )
4121, 38, 40mpjaodan 827 1 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> ( ( A (,) C ) \ ( A (,) B ) ) = ( B [,) C ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,)cico 12177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179  df-ico 12181
This theorem is referenced by:  dya2iocbrsiga  30337  dya2icobrsiga  30338
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