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Theorem iocinif 29543
Description: Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
Assertion
Ref Expression
iocinif |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = if ( A < B , ( B (,] C ) , ( A (,] C ) ) )
Proof of Theorem iocinif
StepHypRef Expression
1 exmid 431 . . 3 |- ( A < B \/ -. A < B )
2 xrltle 11982 . . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> A <_ B ) )
32imp 445 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> A <_ B )
433adantl3 1219 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> A <_ B )
5 iocinioc2 29541 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) )
64, 5syldan 487 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A < B ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) )
76ex 450 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) ) )
87ancld 576 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> ( A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) ) ) )
9 simpl2 1065 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ -. A < B ) -> B e. RR* )
10 simpl1 1064 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ -. A < B ) -> A e. RR* )
11 simpr 477 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ -. A < B ) -> -. A < B )
12 xrlenlt 10103 . . . . . . . . 9 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
1312biimpar 502 . . . . . . . 8 |- ( ( ( B e. RR* /\ A e. RR* ) /\ -. A < B ) -> B <_ A )
149, 10, 11, 13syl21anc 1325 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ -. A < B ) -> B <_ A )
15 3ancoma 1045 . . . . . . . 8 |- ( ( B e. RR* /\ A e. RR* /\ C e. RR* ) <-> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
16 incom 3805 . . . . . . . . 9 |- ( ( B (,] C ) i^i ( A (,] C ) ) = ( ( A (,] C ) i^i ( B (,] C ) )
17 iocinioc2 29541 . . . . . . . . 9 |- ( ( ( B e. RR* /\ A e. RR* /\ C e. RR* ) /\ B <_ A ) -> ( ( B (,] C ) i^i ( A (,] C ) ) = ( A (,] C ) )
1816, 17syl5eqr 2670 . . . . . . . 8 |- ( ( ( B e. RR* /\ A e. RR* /\ C e. RR* ) /\ B <_ A ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) )
1915, 18sylanbr 490 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ B <_ A ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) )
2014, 19syldan 487 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ -. A < B ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) )
2120ex 450 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -. A < B -> ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) ) )
2221ancld 576 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -. A < B -> ( -. A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) ) ) )
238, 22orim12d 883 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B \/ -. A < B ) -> ( ( A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) ) \/ ( -. A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) ) ) ) )
241, 23mpi 20 . 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) ) \/ ( -. A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) ) ) )
25 eqif 4126 . 2 |- ( ( ( A (,] C ) i^i ( B (,] C ) ) = if ( A < B , ( B (,] C ) , ( A (,] C ) ) <-> ( ( A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( B (,] C ) ) \/ ( -. A < B /\ ( ( A (,] C ) i^i ( B (,] C ) ) = ( A (,] C ) ) ) )
2624, 25sylibr 224 1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A (,] C ) i^i ( B (,] C ) ) = if ( A < B , ( B (,] C ) , ( A (,] C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ifcif 4086   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176
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-pre-lttri 10010  ax-pre-lttrn 10011
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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioc 12180
This theorem is referenced by:  pnfneige0  29997
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