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Theorem eleesubd 24612
Description: Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 24611. (Contributed by Scott Fenton, 17-Jul-2013.)
Hypothesis
Ref Expression
eleesubd.1  |-  ( ph  ->  C  =  ( i  e.  ( 1 ... N )  |->  ( ( A `  i )  -  ( B `  i ) ) ) )
Assertion
Ref Expression
eleesubd  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
Distinct variable groups:    i, N    A, i    B, i
Allowed substitution hints:    ph( i)    C( i)

Proof of Theorem eleesubd
StepHypRef Expression
1 eleesubd.1 . . 3  |-  ( ph  ->  C  =  ( i  e.  ( 1 ... N )  |->  ( ( A `  i )  -  ( B `  i ) ) ) )
213ad2ant1 976 . 2  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  C  =  ( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) ) )
3 fveere 24601 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
4 fveere 24601 . . . . . . 7  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
5 resubcl 9127 . . . . . . 7  |-  ( ( ( A `  i
)  e.  RR  /\  ( B `  i )  e.  RR )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
63, 4, 5syl2an 463 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  i  e.  ( 1 ... N ) )  /\  ( B  e.  ( EE `  N )  /\  i  e.  ( 1 ... N
) ) )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
76anandirs 804 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
87ralrimiva 2639 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR )
9 eleenn 24596 . . . . . 6  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
10 mptelee 24595 . . . . . 6  |-  ( N  e.  NN  ->  (
( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )  e.  ( EE `  N )  <->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR ) )
119, 10syl 15 . . . . 5  |-  ( A  e.  ( EE `  N )  ->  (
( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )  e.  ( EE `  N )  <->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR ) )
1211adantr 451 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( ( i  e.  ( 1 ... N
)  |->  ( ( A `
 i )  -  ( B `  i ) ) )  e.  ( EE `  N )  <->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR ) )
138, 12mpbird 223 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )  e.  ( EE `  N ) )
14133adant1 973 . 2  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( i  e.  ( 1 ... N
)  |->  ( ( A `
 i )  -  ( B `  i ) ) )  e.  ( EE `  N ) )
152, 14eqeltrd 2370 1  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    - cmin 9053   NNcn 9762   ...cfz 10798   EEcee 24588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-ee 24591
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