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Theorem i1fmulclem 19057
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 8828 . . . . . . . . . 10  |-  RR  e.  _V
21a1i 10 . . . . . . . . 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 19031 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
7 ffn 5389 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
86, 7syl 15 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
9 eqidd 2284 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
102, 3, 8, 9ofc1 6100 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1110adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  z  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 z )  =  ( A  x.  ( F `  z )
) )
1211adantlr 695 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1312eqeq1d 2291 . . . . 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 2285 . . . . . 6  |-  ( ( F `  z )  =  ( B  /  A )  <->  ( B  /  A )  =  ( F `  z ) )
15 simplr 731 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  RR )
1615recnd 8861 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  CC )
173ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  RR )
1817recnd 8861 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  CC )
196ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F : RR --> RR )
20 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : RR --> RR  /\  z  e.  RR )  ->  ( F `  z
)  e.  RR )
2119, 20sylan 457 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  RR )
2221recnd 8861 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  CC )
23 simpllr 735 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  =/=  0
)
2416, 18, 22, 23divmuld 9558 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( B  /  A )  =  ( F `  z
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2514, 24syl5bb 248 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( F `
 z )  =  ( B  /  A
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2613, 25bitr4d 247 . . . 4  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  z )  =  B  <-> 
( F `  z
)  =  ( B  /  A ) ) )
2726pm5.32da 622 . . 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 ) ) ) )
28 remulcl 8822 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2928adantl 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
30 fconstg 5428 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
313, 30syl 15 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
323snssd 3760 . . . . . . . 8  |-  ( ph  ->  { A }  C_  RR )
33 fss 5397 . . . . . . . 8  |-  ( ( ( RR  X.  { A } ) : RR --> { A }  /\  { A }  C_  RR )  ->  ( RR  X.  { A } ) : RR --> RR )
3431, 32, 33syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
35 inidm 3378 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
3629, 34, 6, 2, 2, 35off 6093 . . . . . 6  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
3736ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
38 ffn 5389 . . . . 5  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
3937, 38syl 15 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
40 fniniseg 5646 . . . 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 ) ) )
4139, 40syl 15 . . 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 ) ) )
4219, 7syl 15 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F  Fn  RR )
43 fniniseg 5646 . . . 4  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( B  /  A ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( B  /  A
) ) ) )
4442, 43syl 15 . . 3  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' F " { ( B  /  A ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( B  /  A
) ) ) )
4527, 41, 443bitr4d 276 . 2  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' ( ( RR  X.  { A } )  o F  x.  F )
" { B }
)  <->  z  e.  ( `' F " { ( B  /  A ) } ) ) )
4645eqrdv 2281 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   {csn 3640    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   RRcr 8736   0cc0 8737    x. cmul 8742    / cdiv 9423   S.1citg1 18970
This theorem is referenced by:  i1fmulc  19058  itg1mulc  19059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-sum 12159  df-itg1 18976
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