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Theorem dvlem 22745
Description: Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
Hypotheses
Ref Expression
dvlem.1  |-  ( ph  ->  F : D --> CC )
dvlem.2  |-  ( ph  ->  D  C_  CC )
dvlem.3  |-  ( ph  ->  B  e.  D )
Assertion
Ref Expression
dvlem  |-  ( (
ph  /\  A  e.  ( D  \  { B } ) )  -> 
( ( ( F `
 A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )

Proof of Theorem dvlem
StepHypRef Expression
1 eldifsn 4119 . 2  |-  ( A  e.  ( D  \  { B } )  <->  ( A  e.  D  /\  A  =/= 
B ) )
2 dvlem.1 . . . . . 6  |-  ( ph  ->  F : D --> CC )
32adantr 466 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  F : D --> CC )
4 simprl 762 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  D )
53, 4ffvelrnd 6029 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  A
)  e.  CC )
6 dvlem.3 . . . . . 6  |-  ( ph  ->  B  e.  D )
76adantr 466 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  D )
83, 7ffvelrnd 6029 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  B
)  e.  CC )
95, 8subcld 9975 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( F `  A )  -  ( F `  B )
)  e.  CC )
10 dvlem.2 . . . . . 6  |-  ( ph  ->  D  C_  CC )
1110adantr 466 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  D  C_  CC )
1211, 4sseldd 3462 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  CC )
1311, 7sseldd 3462 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  CC )
1412, 13subcld 9975 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  e.  CC )
15 simprr 764 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  =/=  B )
1612, 13, 15subne0d 9984 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  =/=  0 )
179, 14, 16divcld 10372 . 2  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( ( F `
 A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
181, 17sylan2b 477 1  |-  ( (
ph  /\  A  e.  ( D  \  { B } ) )  -> 
( ( ( F `
 A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1867    =/= wne 2616    \ cdif 3430    C_ wss 3433   {csn 3993   -->wf 5588   ` cfv 5592  (class class class)co 6296   CCcc 9526    - cmin 9849    / cdiv 10258
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-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  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-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-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
This theorem is referenced by:  perfdvf  22752  dvreslem  22758  dvcnp  22767  dvcnp2  22768  dvaddbr  22786  dvmulbr  22787  dvcobr  22794  dvcjbr  22797  dvcnvlem  22822  dvferm1  22831  dvferm2  22833  ftc1lem6  22887  ulmdvlem3  23248  ftc1cnnc  31749  fperdvper  37395
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