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Theorem cpnres 19390
Description: The restriction of a  C ^n function is  C ^n. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
cpnres  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( F  |`  S )  e.  ( ( C ^n `  S ) `
 N ) )

Proof of Theorem cpnres
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  S  e.  { RR ,  CC } )
2 simpr 447 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  F  e.  ( (
C ^n `  CC ) `  N )
)
3 ssid 3273 . . . . . 6  |-  CC  C_  CC
4 elfvdm 5637 . . . . . . . 8  |-  ( F  e.  ( ( C ^n `  CC ) `
 N )  ->  N  e.  dom  ( C ^n `  CC ) )
54adantl 452 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  N  e.  dom  ( C ^n `  CC ) )
6 fncpn 19386 . . . . . . . . 9  |-  ( CC  C_  CC  ->  ( C ^n `  CC )  Fn 
NN0 )
73, 6ax-mp 8 . . . . . . . 8  |-  ( C ^n `  CC )  Fn  NN0
8 fndm 5425 . . . . . . . 8  |-  ( ( C ^n `  CC )  Fn  NN0  ->  dom  ( C ^n `  CC )  =  NN0 )
97, 8mp1i 11 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  dom  ( C ^n `  CC )  =  NN0 )
105, 9eleqtrd 2434 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  N  e.  NN0 )
11 elcpn 19387 . . . . . 6  |-  ( ( CC  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C ^n `  CC ) `  N )  <->  ( F  e.  ( CC 
^pm  CC )  /\  (
( CC  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) ) )
123, 10, 11sylancr 644 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( F  e.  ( ( C ^n `  CC ) `  N )  <-> 
( F  e.  ( CC  ^pm  CC )  /\  ( ( CC  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) ) )
132, 12mpbid 201 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( F  e.  ( CC  ^pm  CC )  /\  ( ( CC  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) )
1413simpld 445 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  F  e.  ( CC  ^pm 
CC ) )
15 pmresg 6883 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
161, 14, 15syl2anc 642 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( F  |`  S )  e.  ( CC  ^pm  S ) )
1713simprd 449 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( CC  D n F ) `  N
)  e.  ( dom 
F -cn-> CC ) )
18 cncff 18500 . . . . . 6  |-  ( ( ( CC  D n F ) `  N
)  e.  ( dom 
F -cn-> CC )  ->  (
( CC  D n F ) `  N
) : dom  F --> CC )
1917, 18syl 15 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( CC  D n F ) `  N
) : dom  F --> CC )
20 fdm 5476 . . . . 5  |-  ( ( ( CC  D n F ) `  N
) : dom  F --> CC  ->  dom  ( ( CC  D n F ) `
 N )  =  dom  F )
2119, 20syl 15 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  dom  ( ( CC  D n F ) `  N
)  =  dom  F
)
22 dvnres 19384 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC  D n F ) `  N
)  =  dom  F
)  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) )
231, 14, 10, 21, 22syl31anc 1185 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) )
24 resres 5050 . . . . . . 7  |-  ( ( ( ( CC  D n F ) `  N
)  |`  S )  |`  dom  F )  =  ( ( ( CC  D n F ) `  N
)  |`  ( S  i^i  dom 
F ) )
25 rescom 5062 . . . . . . 7  |-  ( ( ( ( CC  D n F ) `  N
)  |`  S )  |`  dom  F )  =  ( ( ( ( CC  D n F ) `
 N )  |`  dom  F )  |`  S )
2624, 25eqtr3i 2380 . . . . . 6  |-  ( ( ( CC  D n F ) `  N
)  |`  ( S  i^i  dom 
F ) )  =  ( ( ( ( CC  D n F ) `  N )  |`  dom  F )  |`  S )
27 ffn 5472 . . . . . . . 8  |-  ( ( ( CC  D n F ) `  N
) : dom  F --> CC  ->  ( ( CC  D n F ) `
 N )  Fn 
dom  F )
28 fnresdm 5435 . . . . . . . 8  |-  ( ( ( CC  D n F ) `  N
)  Fn  dom  F  ->  ( ( ( CC  D n F ) `
 N )  |`  dom  F )  =  ( ( CC  D n F ) `  N
) )
2919, 27, 283syl 18 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( ( CC  D n F ) `
 N )  |`  dom  F )  =  ( ( CC  D n F ) `  N
) )
3029reseq1d 5036 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( ( ( CC  D n F ) `  N )  |`  dom  F )  |`  S )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) )
3126, 30syl5eq 2402 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( ( CC  D n F ) `
 N )  |`  ( S  i^i  dom  F
) )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) )
32 inss2 3466 . . . . . 6  |-  ( S  i^i  dom  F )  C_ 
dom  F
33 rescncf 18504 . . . . . 6  |-  ( ( S  i^i  dom  F
)  C_  dom  F  -> 
( ( ( CC  D n F ) `
 N )  e.  ( dom  F -cn-> CC )  ->  ( (
( CC  D n F ) `  N
)  |`  ( S  i^i  dom 
F ) )  e.  ( ( S  i^i  dom 
F ) -cn-> CC ) ) )
3432, 17, 33mpsyl 59 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( ( CC  D n F ) `
 N )  |`  ( S  i^i  dom  F
) )  e.  ( ( S  i^i  dom  F ) -cn-> CC ) )
3531, 34eqeltrrd 2433 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( ( CC  D n F ) `
 N )  |`  S )  e.  ( ( S  i^i  dom  F ) -cn-> CC ) )
36 dmres 5058 . . . . 5  |-  dom  ( F  |`  S )  =  ( S  i^i  dom  F )
3736oveq1i 5955 . . . 4  |-  ( dom  ( F  |`  S )
-cn-> CC )  =  ( ( S  i^i  dom  F ) -cn-> CC )
3835, 37syl6eleqr 2449 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( ( CC  D n F ) `
 N )  |`  S )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
3923, 38eqeltrd 2432 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( S  D n ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
40 recnprss 19358 . . . 4  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
4140adantr 451 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  ->  S  C_  CC )
42 elcpn 19387 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  (
( F  |`  S )  e.  ( ( C ^n `  S ) `
 N )  <->  ( ( F  |`  S )  e.  ( CC  ^pm  S
)  /\  ( ( S  D n ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) ) ) )
4341, 10, 42syl2anc 642 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( ( F  |`  S )  e.  ( ( C ^n `  S ) `  N
)  <->  ( ( F  |`  S )  e.  ( CC  ^pm  S )  /\  ( ( S  D n ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) ) ) )
4416, 39, 43mpbir2and 888 1  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n
`  CC ) `  N ) )  -> 
( F  |`  S )  e.  ( ( C ^n `  S ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   {cpr 3717   dom cdm 4771    |` cres 4773    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945    ^pm cpm 6861   CCcc 8825   RRcr 8826   NN0cn0 10057   -cn->ccncf 18483    D ncdvn 19318   C ^nccpn 19319
This theorem is referenced by:  aalioulem3  19818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-icc 10755  df-fz 10875  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-plusg 13318  df-mulr 13319  df-starv 13320  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-rest 13426  df-topn 13427  df-topgen 13443  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cnp 17064  df-haus 17149  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-cncf 18485  df-limc 19320  df-dv 19321  df-dvn 19322  df-cpn 19323
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