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Theorem difico 26211
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 11520 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  =  (/) )
2 undif4 3836 . . . 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 3848 . . . . 5  |-  ( ( B [,) C ) 
\  ( B [,) C ) )  =  (/)
65uneq2i 3608 . . . 4  |-  ( ( A [,) B )  u.  ( ( B [,) C )  \ 
( B [,) C
) ) )  =  ( ( A [,) B )  u.  (/) )
7 un0 3763 . . . 4  |-  ( ( A [,) B )  u.  (/) )  =  ( A [,) B )
86, 7eqtri 2480 . . 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 11519 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  -> 
( ( A [,) B )  u.  ( B [,) C ) )  =  ( A [,) C ) )
1110difeq1d 3574 . 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 2501 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 965    = wceq 1370    e. wcel 1758    \ cdif 3426    u. cun 3427    i^i cin 3428   (/)c0 3738   class class class wbr 4393  (class class class)co 6193   RR*cxr 9521    <_ cle 9523   [,)cico 11406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-pre-lttri 9460  ax-pre-lttrn 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-ico 11410
This theorem is referenced by:  sxbrsigalem2  26838
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