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Theorem dvlem 22063
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 4152 . 2  |-  ( A  e.  ( D  \  { B } )  <->  ( A  e.  D  /\  A  =/= 
B ) )
2 dvlem.1 . . . . . 6  |-  ( ph  ->  F : D --> CC )
32adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  F : D --> CC )
4 simprl 755 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  D )
53, 4ffvelrnd 6022 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  A
)  e.  CC )
6 dvlem.3 . . . . . 6  |-  ( ph  ->  B  e.  D )
76adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  D )
83, 7ffvelrnd 6022 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  B
)  e.  CC )
95, 8subcld 9930 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( F `  A )  -  ( F `  B )
)  e.  CC )
10 dvlem.2 . . . . . 6  |-  ( ph  ->  D  C_  CC )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  D  C_  CC )
1211, 4sseldd 3505 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  CC )
1311, 7sseldd 3505 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  CC )
1412, 13subcld 9930 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  e.  CC )
15 simprr 756 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  =/=  B )
1612, 13, 15subne0d 9939 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  =/=  0 )
179, 14, 16divcld 10320 . 2  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( ( F `
 A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
181, 17sylan2b 475 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 369    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   {csn 4027   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490    - cmin 9805    / cdiv 10206
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-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 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207
This theorem is referenced by:  perfdvf  22070  dvreslem  22076  dvcnp  22085  dvcnp2  22086  dvaddbr  22104  dvmulbr  22105  dvcobr  22112  dvcjbr  22115  dvcnvlem  22140  dvferm1  22149  dvferm2  22151  ftc1lem6  22205  ulmdvlem3  22559  ftc1cnnc  29694  fperdvper  31276
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