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Theorem i1fmulclem 21936
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 9584 . . . . . . . . . 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 21910 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
7 ffn 5731 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  RR )
9 eqidd 2468 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
102, 3, 8, 9ofc1 6548 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1110adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  z  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 z )  =  ( A  x.  ( F `  z )
) )
1211adantlr 714 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  ( A  x.  ( F `
 z ) ) )
1312eqeq1d 2469 . . . . 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 2476 . . . . . 6  |-  ( ( F `  z )  =  ( B  /  A )  <->  ( B  /  A )  =  ( F `  z ) )
15 simplr 754 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  RR )
1615recnd 9623 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  B  e.  CC )
173ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  RR )
1817recnd 9623 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  e.  CC )
196ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F : RR --> RR )
2019ffvelrnda 6022 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  RR )
2120recnd 9623 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( F `  z )  e.  CC )
22 simpllr 758 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  A  =/=  0
)
2316, 18, 21, 22divmuld 10343 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( B  /  A )  =  ( F `  z
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2414, 23syl5bb 257 . . . . 5  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( F `
 z )  =  ( B  /  A
)  <->  ( A  x.  ( F `  z ) )  =  B ) )
2513, 24bitr4d 256 . . . 4  |-  ( ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  /\  z  e.  RR )  ->  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  z )  =  B  <-> 
( F `  z
)  =  ( B  /  A ) ) )
2625pm5.32da 641 . . 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 9578 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2827adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
29 fconstg 5772 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( RR  X.  { A }
) : RR --> { A } )
303, 29syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
313snssd 4172 . . . . . . . 8  |-  ( ph  ->  { A }  C_  RR )
32 fss 5739 . . . . . . . 8  |-  ( ( ( RR  X.  { A } ) : RR --> { A }  /\  { A }  C_  RR )  ->  ( RR  X.  { A } ) : RR --> RR )
3330, 31, 32syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> RR )
34 inidm 3707 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
3528, 33, 6, 2, 2, 34off 6539 . . . . . 6  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
3635ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
37 ffn 5731 . . . . 5  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
3836, 37syl 16 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
39 fniniseg 6003 . . . 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 ) ) )
4038, 39syl 16 . . 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 ) ) )
4119, 7syl 16 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  F  Fn  RR )
42 fniniseg 6003 . . . 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 285 . 2  |-  ( ( ( ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  (
z  e.  ( `' ( ( RR  X.  { A } )  oF  x.  F )
" { B }
)  <->  z  e.  ( `' F " { ( B  /  A ) } ) ) )
4544eqrdv 2464 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 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    C_ wss 3476   {csn 4027    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    oFcof 6523   RRcr 9492   0cc0 9493    x. cmul 9498    / cdiv 10207   S.1citg1 21851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-sum 13475  df-itg1 21856
This theorem is referenced by:  i1fmulc  21937  itg1mulc  21938
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