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Theorem dvlem 21386
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 4015 . 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 5859 . . . 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 5859 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  B
)  e.  CC )
95, 8subcld 9734 . . 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 3372 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  CC )
1311, 7sseldd 3372 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  CC )
1412, 13subcld 9734 . . 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 9743 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  =/=  0 )
179, 14, 16divcld 10122 . 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 1756    =/= wne 2620    \ cdif 3340    C_ wss 3343   {csn 3892   -->wf 5429   ` cfv 5433  (class class class)co 6106   CCcc 9295    - cmin 9610    / cdiv 10008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009
This theorem is referenced by:  perfdvf  21393  dvreslem  21399  dvcnp  21408  dvcnp2  21409  dvaddbr  21427  dvmulbr  21428  dvcobr  21435  dvcjbr  21438  dvcnvlem  21463  dvferm1  21472  dvferm2  21474  ftc1lem6  21528  ulmdvlem3  21882  ftc1cnnc  28485
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