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Theorem dvlem 19246
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 3749 . 2  |-  ( A  e.  ( D  \  { B } )  <->  ( A  e.  D  /\  A  =/= 
B ) )
2 dvlem.1 . . . . . 6  |-  ( ph  ->  F : D --> CC )
32adantr 451 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  F : D --> CC )
4 simprl 732 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  D )
53, 4ffvelrnd 5666 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  A
)  e.  CC )
6 dvlem.3 . . . . . 6  |-  ( ph  ->  B  e.  D )
76adantr 451 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  D )
83, 7ffvelrnd 5666 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( F `  B
)  e.  CC )
95, 8subcld 9157 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( F `  A )  -  ( F `  B )
)  e.  CC )
10 dvlem.2 . . . . . 6  |-  ( ph  ->  D  C_  CC )
1110adantr 451 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  D  C_  CC )
1211, 4sseldd 3181 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  e.  CC )
1311, 7sseldd 3181 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  B  e.  CC )
1412, 13subcld 9157 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  e.  CC )
15 simprr 733 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  ->  A  =/=  B )
16 subeq0 9073 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  0  <-> 
A  =  B ) )
1712, 13, 16syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( A  -  B )  =  0  <-> 
A  =  B ) )
1817necon3bid 2481 . . . 4  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( A  -  B )  =/=  0  <->  A  =/=  B ) )
1915, 18mpbird 223 . . 3  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( A  -  B
)  =/=  0 )
209, 14, 19divcld 9536 . 2  |-  ( (
ph  /\  ( A  e.  D  /\  A  =/= 
B ) )  -> 
( ( ( F `
 A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
211, 20sylan2b 461 1  |-  ( (
ph  /\  A  e.  ( D  \  { B } ) )  -> 
( ( ( F `
 A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    - cmin 9037    / cdiv 9423
This theorem is referenced by:  perfdvf  19253  dvreslem  19259  dvcnp  19268  dvcnp2  19269  dvaddbr  19287  dvmulbr  19288  dvcobr  19295  dvcjbr  19298  dvcnvlem  19323  dvferm1  19332  dvferm2  19334  ftc1lem6  19388  ulmdvlem3  19779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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