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Theorem dvaddf 22514
Description: The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
Hypotheses
Ref Expression
dvaddf.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
dvaddf.f  |-  ( ph  ->  F : X --> CC )
dvaddf.g  |-  ( ph  ->  G : X --> CC )
dvaddf.df  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
dvaddf.dg  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
Assertion
Ref Expression
dvaddf  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) )  =  ( ( S  _D  F
)  oF  +  ( S  _D  G
) ) )

Proof of Theorem dvaddf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvaddf.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 dvaddf.df . . . . 5  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
3 dvbsss 22475 . . . . 5  |-  dom  ( S  _D  F )  C_  S
42, 3syl6eqssr 3540 . . . 4  |-  ( ph  ->  X  C_  S )
51, 4ssexd 4584 . . 3  |-  ( ph  ->  X  e.  _V )
6 dvfg 22479 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
71, 6syl 16 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
82feq2d 5700 . . . . 5  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
97, 8mpbid 210 . . . 4  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
10 ffn 5713 . . . 4  |-  ( ( S  _D  F ) : X --> CC  ->  ( S  _D  F )  Fn  X )
119, 10syl 16 . . 3  |-  ( ph  ->  ( S  _D  F
)  Fn  X )
12 dvfg 22479 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  G ) : dom  ( S  _D  G
) --> CC )
131, 12syl 16 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
14 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1514feq2d 5700 . . . . 5  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
1613, 15mpbid 210 . . . 4  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
17 ffn 5713 . . . 4  |-  ( ( S  _D  G ) : X --> CC  ->  ( S  _D  G )  Fn  X )
1816, 17syl 16 . . 3  |-  ( ph  ->  ( S  _D  G
)  Fn  X )
19 dvfg 22479 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  oF  +  G ) ) : dom  ( S  _D  ( F  oF  +  G )
) --> CC )
201, 19syl 16 . . . . 5  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : dom  ( S  _D  ( F  oF  +  G
) ) --> CC )
21 recnprss 22477 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
221, 21syl 16 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
23 addcl 9563 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
2423adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
25 dvaddf.f . . . . . . . . 9  |-  ( ph  ->  F : X --> CC )
26 dvaddf.g . . . . . . . . 9  |-  ( ph  ->  G : X --> CC )
27 inidm 3693 . . . . . . . . 9  |-  ( X  i^i  X )  =  X
2824, 25, 26, 5, 5, 27off 6527 . . . . . . . 8  |-  ( ph  ->  ( F  oF  +  G ) : X --> CC )
2922, 28, 4dvbss 22474 . . . . . . 7  |-  ( ph  ->  dom  ( S  _D  ( F  oF  +  G ) )  C_  X )
3025adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
314adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
3226adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
3322adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
34 fvex 5858 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3534a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
36 fvex 5858 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3736a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
382eleq2d 2524 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
3938biimpar 483 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
401adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
41 ffun 5715 . . . . . . . . . . . . 13  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
42 funfvbrb 5976 . . . . . . . . . . . . 13  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4340, 6, 41, 424syl 21 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4439, 43mpbid 210 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
4514eleq2d 2524 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
4645biimpar 483 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
47 ffun 5715 . . . . . . . . . . . . 13  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
48 funfvbrb 5976 . . . . . . . . . . . . 13  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4940, 12, 47, 484syl 21 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5046, 49mpbid 210 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
51 eqid 2454 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
5230, 31, 32, 31, 33, 35, 37, 44, 50, 51dvaddbr 22510 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  ( F  oF  +  G
) ) ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) ) )
53 reldv 22443 . . . . . . . . . . 11  |-  Rel  ( S  _D  ( F  oF  +  G )
)
5453releldmi 5228 . . . . . . . . . 10  |-  ( x ( S  _D  ( F  oF  +  G
) ) ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) )  ->  x  e.  dom  ( S  _D  ( F  oF  +  G )
) )
5552, 54syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  oF  +  G ) ) )
5655ex 432 . . . . . . . 8  |-  ( ph  ->  ( x  e.  X  ->  x  e.  dom  ( S  _D  ( F  oF  +  G )
) ) )
5756ssrdv 3495 . . . . . . 7  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  oF  +  G )
) )
5829, 57eqssd 3506 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  +  G ) )  =  X )
5958feq2d 5700 . . . . 5  |-  ( ph  ->  ( ( S  _D  ( F  oF  +  G ) ) : dom  ( S  _D  ( F  oF  +  G ) ) --> CC  <->  ( S  _D  ( F  oF  +  G
) ) : X --> CC ) )
6020, 59mpbid 210 . . . 4  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : X --> CC )
61 ffn 5713 . . . 4  |-  ( ( S  _D  ( F  oF  +  G
) ) : X --> CC  ->  ( S  _D  ( F  oF  +  G ) )  Fn  X )
6260, 61syl 16 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) )  Fn  X
)
63 eqidd 2455 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
64 eqidd 2455 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  =  ( ( S  _D  G ) `  x ) )
6530, 31, 32, 31, 40, 39, 46dvadd 22512 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  ( F  oF  +  G
) ) `  x
)  =  ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) ) )
6665eqcomd 2462 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  +  ( ( S  _D  G ) `
 x ) )  =  ( ( S  _D  ( F  oF  +  G )
) `  x )
)
675, 11, 18, 62, 63, 64, 66offveq 6534 . 2  |-  ( ph  ->  ( ( S  _D  F )  oF  +  ( S  _D  G ) )  =  ( S  _D  ( F  oF  +  G
) ) )
6867eqcomd 2462 1  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) )  =  ( ( S  _D  F
)  oF  +  ( S  _D  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   {cpr 4018   class class class wbr 4439   dom cdm 4988   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   CCcc 9479   RRcr 9480    + caddc 9484   TopOpenctopn 14914  ℂfldccnfld 18618    _D cdv 22436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-limc 22439  df-dv 22440
This theorem is referenced by:  dvmptadd  22532
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