Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cncfioobdlem Structured version   Visualization version   Unicode version

Theorem cncfioobdlem 37774
Description:  G actually extends  F. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
cncfioobdlem.a  |-  ( ph  ->  A  e.  RR )
cncfioobdlem.b  |-  ( ph  ->  B  e.  RR )
cncfioobdlem.f  |-  ( ph  ->  F : ( A (,) B ) --> V )
cncfioobdlem.g  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
cncfioobdlem.c  |-  ( ph  ->  C  e.  ( A (,) B ) )
Assertion
Ref Expression
cncfioobdlem  |-  ( ph  ->  ( G `  C
)  =  ( F `
 C ) )
Distinct variable groups:    x, A    x, B    x, C    x, F    ph, x
Allowed substitution hints:    R( x)    G( x)    L( x)    V( x)

Proof of Theorem cncfioobdlem
StepHypRef Expression
1 cncfioobdlem.g . . 3  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
21a1i 11 . 2  |-  ( ph  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) ) )
3 cncfioobdlem.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
43adantr 467 . . . . . 6  |-  ( (
ph  /\  x  =  C )  ->  A  e.  RR )
5 cncfioobdlem.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  ( A (,) B ) )
63rexrd 9690 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
7 cncfioobdlem.b . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  RR )
87rexrd 9690 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
9 elioo2 11677 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <  B ) ) )
106, 8, 9syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( C  e.  ( A (,) B )  <-> 
( C  e.  RR  /\  A  <  C  /\  C  <  B ) ) )
115, 10mpbid 214 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  RR  /\  A  <  C  /\  C  <  B ) )
1211simp2d 1021 . . . . . . . 8  |-  ( ph  ->  A  <  C )
1312adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  =  C )  ->  A  <  C )
14 eqcom 2458 . . . . . . . . 9  |-  ( x  =  C  <->  C  =  x )
1514biimpi 198 . . . . . . . 8  |-  ( x  =  C  ->  C  =  x )
1615adantl 468 . . . . . . 7  |-  ( (
ph  /\  x  =  C )  ->  C  =  x )
1713, 16breqtrd 4427 . . . . . 6  |-  ( (
ph  /\  x  =  C )  ->  A  <  x )
184, 17gtned 9770 . . . . 5  |-  ( (
ph  /\  x  =  C )  ->  x  =/=  A )
1918neneqd 2629 . . . 4  |-  ( (
ph  /\  x  =  C )  ->  -.  x  =  A )
2019iffalsed 3892 . . 3  |-  ( (
ph  /\  x  =  C )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
21 simpr 463 . . . . . . 7  |-  ( (
ph  /\  x  =  C )  ->  x  =  C )
2211simp1d 1020 . . . . . . . 8  |-  ( ph  ->  C  e.  RR )
2322adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  =  C )  ->  C  e.  RR )
2421, 23eqeltrd 2529 . . . . . 6  |-  ( (
ph  /\  x  =  C )  ->  x  e.  RR )
2511simp3d 1022 . . . . . . . 8  |-  ( ph  ->  C  <  B )
2625adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  =  C )  ->  C  <  B )
2721, 26eqbrtrd 4423 . . . . . 6  |-  ( (
ph  /\  x  =  C )  ->  x  <  B )
2824, 27ltned 9771 . . . . 5  |-  ( (
ph  /\  x  =  C )  ->  x  =/=  B )
2928neneqd 2629 . . . 4  |-  ( (
ph  /\  x  =  C )  ->  -.  x  =  B )
3029iffalsed 3892 . . 3  |-  ( (
ph  /\  x  =  C )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  ( F `  x ) )
3121fveq2d 5869 . . 3  |-  ( (
ph  /\  x  =  C )  ->  ( F `  x )  =  ( F `  C ) )
3220, 30, 313eqtrd 2489 . 2  |-  ( (
ph  /\  x  =  C )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  C ) )
33 ioossicc 11720 . . 3  |-  ( A (,) B )  C_  ( A [,] B )
3433, 5sseldi 3430 . 2  |-  ( ph  ->  C  e.  ( A [,] B ) )
35 cncfioobdlem.f . . 3  |-  ( ph  ->  F : ( A (,) B ) --> V )
3635, 5ffvelrnd 6023 . 2  |-  ( ph  ->  ( F `  C
)  e.  V )
372, 32, 34, 36fvmptd 5954 1  |-  ( ph  ->  ( G `  C
)  =  ( F `
 C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   RR*cxr 9674    < clt 9675   (,)cioo 11635   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-ioo 11639  df-icc 11642
This theorem is referenced by:  cncfioobd  37775
  Copyright terms: Public domain W3C validator