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Theorem i1fmulclem 22567
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 }
)  oF  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 9619 . . . . . . . . . 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 22541 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
7 ffn 5737 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
86, 7syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
9 eqidd 2421 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
102, 3, 8, 9ofc1 6559 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1110adantlr 719 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  z  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 z )  =  ( A  x.  ( F `  z )
) )
1211adantlr 719 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1312eqeq1d 2422 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  B  <-> 
( A  x.  ( F `  z )
)  =  B ) )
14 eqcom 2429 . . . . . 6  |-  ( ( F `  z )  =  ( B  /  A )  <->  ( B  /  A )  =  ( F `  z ) )
15 simplr 760 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  RR )
1615recnd 9658 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  CC )
173ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  RR )
1817recnd 9658 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  CC )
196ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F : RR --> RR )
2019ffvelrnda 6028 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  RR )
2120recnd 9658 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  CC )
22 simpllr 767 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  =/=  0
)
2316, 18, 21, 22divmuld 10394 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( B  /  A )  =  ( F `  z
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2414, 23syl5bb 260 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( F `
 z )  =  ( B  /  A
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2513, 24bitr4d 259 . . . 4  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  B  <-> 
( F `  z
)  =  ( B  /  A ) ) )
2625pm5.32da 645 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( z  e.  RR  /\  ( ( ( RR 
X.  { A }
)  oF  x.  F ) `  z
)  =  B )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  ( B  /  A ) ) ) )
27 remulcl 9613 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2827adantl 467 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
29 fconstg 5778 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
303, 29syl 17 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
313snssd 4139 . . . . . . . 8  |-  ( ph  ->  { A }  C_  RR )
3230, 31fssd 5746 . . . . . . 7  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
33 inidm 3668 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
3428, 32, 6, 2, 2, 33off 6551 . . . . . 6  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
3534ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
36 ffn 5737 . . . . 5  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
3735, 36syl 17 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
38 fniniseg 6009 . . . 4  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  Fn  RR  ->  ( z  e.  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { B } )  <->  ( z  e.  RR  /\  ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  B ) ) )
3937, 38syl 17 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' ( ( RR  X.  { A } )  oF  x.  F )
" { B }
)  <->  ( z  e.  RR  /\  ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  B ) ) )
4019, 7syl 17 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F  Fn  RR )
41 fniniseg 6009 . . . 4  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( B  /  A ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( B  /  A
) ) ) )
4240, 41syl 17 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' F " { ( B  /  A ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( B  /  A
) ) ) )
4326, 39, 423bitr4d 288 . 2  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' ( ( RR  X.  { A } )  oF  x.  F )
" { B }
)  <->  z  e.  ( `' F " { ( B  /  A ) } ) ) )
4443eqrdv 2417 1  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " { B } )  =  ( `' F " { ( B  /  A ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078   {csn 3993    X. cxp 4843   `'ccnv 4844   dom cdm 4845   "cima 4848    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296    oFcof 6534   RRcr 9527   0cc0 9528    x. cmul 9533    / cdiv 10258   S.1citg1 22480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-sum 13720  df-itg1 22485
This theorem is referenced by:  i1fmulc  22568  itg1mulc  22569
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