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Theorem i1fmulclem 19586
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 9073 . . . . . . . . . 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 19560 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
7 ffn 5583 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
9 eqidd 2436 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
102, 3, 8, 9ofc1 6319 . . . . . . . 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 2443 . . . . 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 2437 . . . . . 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 9106 . . . . . . 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 9106 . . . . . . 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 5862 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  RR )
2120recnd 9106 . . . . . . 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 9804 . . . . . 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 9067 . . . . . . . 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 5622 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
303, 29syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
313snssd 3935 . . . . . . . 8  |-  ( ph  ->  { A }  C_  RR )
32 fss 5591 . . . . . . . 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 3542 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
3528, 33, 6, 2, 2, 34off 6312 . . . . . 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 5583 . . . . 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 5843 . . . 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 5843 . . . 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 2433 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 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   {csn 3806    X. cxp 4868   `'ccnv 4869   dom cdm 4870   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   RRcr 8981   0cc0 8982    x. cmul 8987    / cdiv 9669   S.1citg1 19499
This theorem is referenced by:  i1fmulc  19587  itg1mulc  19588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-sum 12472  df-itg1 19505
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