Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  difico Structured version   Unicode version

Theorem difico 27248
Description: The difference between two closed-below, open-above intervals sharing the same upper bound (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
difico  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( A [,) C )  \  ( B [,) C ) )  =  ( A [,) B ) )

Proof of Theorem difico
StepHypRef Expression
1 icodisj 11634 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  =  (/) )
2 undif4 3876 . . . 4  |-  ( ( ( A [,) B
)  i^i  ( B [,) C ) )  =  (/)  ->  ( ( A [,) B )  u.  ( ( B [,) C )  \  ( B [,) C ) ) )  =  ( ( ( A [,) B
)  u.  ( B [,) C ) ) 
\  ( B [,) C ) ) )
31, 2syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  u.  ( ( B [,) C ) 
\  ( B [,) C ) ) )  =  ( ( ( A [,) B )  u.  ( B [,) C ) )  \ 
( B [,) C
) ) )
43adantr 465 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( A [,) B )  u.  (
( B [,) C
)  \  ( B [,) C ) ) )  =  ( ( ( A [,) B )  u.  ( B [,) C ) )  \ 
( B [,) C
) ) )
5 difid 3888 . . . . 5  |-  ( ( B [,) C ) 
\  ( B [,) C ) )  =  (/)
65uneq2i 3648 . . . 4  |-  ( ( A [,) B )  u.  ( ( B [,) C )  \ 
( B [,) C
) ) )  =  ( ( A [,) B )  u.  (/) )
7 un0 3803 . . . 4  |-  ( ( A [,) B )  u.  (/) )  =  ( A [,) B )
86, 7eqtri 2489 . . 3  |-  ( ( A [,) B )  u.  ( ( B [,) C )  \ 
( B [,) C
) ) )  =  ( A [,) B
)
98a1i 11 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( A [,) B )  u.  (
( B [,) C
)  \  ( B [,) C ) ) )  =  ( A [,) B ) )
10 icoun 11633 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( A [,) B )  u.  ( B [,) C ) )  =  ( A [,) C ) )
1110difeq1d 3614 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( ( A [,) B )  u.  ( B [,) C
) )  \  ( B [,) C ) )  =  ( ( A [,) C )  \ 
( B [,) C
) ) )
124, 9, 113eqtr3rd 2510 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( A [,) C )  \  ( B [,) C ) )  =  ( A [,) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    \ cdif 3466    u. cun 3467    i^i cin 3468   (/)c0 3778   class class class wbr 4440  (class class class)co 6275   RR*cxr 9616    <_ cle 9618   [,)cico 11520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ico 11524
This theorem is referenced by:  sxbrsigalem2  27883
  Copyright terms: Public domain W3C validator