MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fmulclem Unicode version

Theorem i1fmulclem 19463
Description: Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
i1fmulc.2  |-  ( ph  ->  F  e.  dom  S.1 )
i1fmulc.3  |-  ( ph  ->  A  e.  RR )
Assertion
Ref Expression
i1fmulclem  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " { B } )  =  ( `' F " { ( B  /  A ) } ) )

Proof of Theorem i1fmulclem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 9016 . . . . . . . . . 10  |-  RR  e.  _V
21a1i 11 . . . . . . . . 9  |-  ( ph  ->  RR  e.  _V )
3 i1fmulc.3 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
4 i1fmulc.2 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  dom  S.1 )
5 i1ff 19437 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
7 ffn 5533 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
9 eqidd 2390 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
102, 3, 8, 9ofc1 6268 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1110adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  z  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 z )  =  ( A  x.  ( F `  z )
) )
1211adantlr 696 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1312eqeq1d 2397 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  B  <-> 
( A  x.  ( F `  z )
)  =  B ) )
14 eqcom 2391 . . . . . 6  |-  ( ( F `  z )  =  ( B  /  A )  <->  ( B  /  A )  =  ( F `  z ) )
15 simplr 732 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  RR )
1615recnd 9049 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  CC )
173ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  RR )
1817recnd 9049 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  CC )
196ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F : RR --> RR )
2019ffvelrnda 5811 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  RR )
2120recnd 9049 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  CC )
22 simpllr 736 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  =/=  0
)
2316, 18, 21, 22divmuld 9746 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( B  /  A )  =  ( F `  z
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2414, 23syl5bb 249 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( F `
 z )  =  ( B  /  A
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2513, 24bitr4d 248 . . . 4  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  B  <-> 
( F `  z
)  =  ( B  /  A ) ) )
2625pm5.32da 623 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( z  e.  RR  /\  ( ( ( RR 
X.  { A }
)  o F  x.  F ) `  z
)  =  B )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  ( B  /  A ) ) ) )
27 remulcl 9010 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2827adantl 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
29 fconstg 5572 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
303, 29syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
313snssd 3888 . . . . . . . 8  |-  ( ph  ->  { A }  C_  RR )
32 fss 5541 . . . . . . . 8  |-  ( ( ( RR  X.  { A } ) : RR --> { A }  /\  { A }  C_  RR )  ->  ( RR  X.  { A } ) : RR --> RR )
3330, 31, 32syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
34 inidm 3495 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
3528, 33, 6, 2, 2, 34off 6261 . . . . . 6  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
3635ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
37 ffn 5533 . . . . 5  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
3836, 37syl 16 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
39 fniniseg 5792 . . . 4  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  Fn  RR  ->  ( z  e.  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { B } )  <->  ( z  e.  RR  /\  ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  B ) ) )
4038, 39syl 16 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' ( ( RR  X.  { A } )  o F  x.  F )
" { B }
)  <->  ( z  e.  RR  /\  ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  B ) ) )
4119, 7syl 16 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F  Fn  RR )
42 fniniseg 5792 . . . 4  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( B  /  A ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( B  /  A
) ) ) )
4341, 42syl 16 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' F " { ( B  /  A ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( B  /  A
) ) ) )
4426, 40, 433bitr4d 277 . 2  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' ( ( RR  X.  { A } )  o F  x.  F )
" { B }
)  <->  z  e.  ( `' F " { ( B  /  A ) } ) ) )
4544eqrdv 2387 1  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " { B } )  =  ( `' F " { ( B  /  A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   _Vcvv 2901    C_ wss 3265   {csn 3759    X. cxp 4818   `'ccnv 4819   dom cdm 4820   "cima 4823    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022    o Fcof 6244   RRcr 8924   0cc0 8925    x. cmul 8930    / cdiv 9611   S.1citg1 19376
This theorem is referenced by:  i1fmulc  19464  itg1mulc  19465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-sum 12409  df-itg1 19382
  Copyright terms: Public domain W3C validator