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Theorem eleesubd 23311
Description: Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 23310. (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 1009 . 2  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  C  =  ( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) ) )
3 fveere 23300 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
4 fveere 23300 . . . . . . 7  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
5 resubcl 9785 . . . . . . 7  |-  ( ( ( A `  i
)  e.  RR  /\  ( B `  i )  e.  RR )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
63, 4, 5syl2an 477 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  i  e.  ( 1 ... N ) )  /\  ( B  e.  ( EE `  N )  /\  i  e.  ( 1 ... N
) ) )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
76anandirs 827 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
87ralrimiva 2830 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR )
9 eleenn 23295 . . . . . 6  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
10 mptelee 23294 . . . . . 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 16 . . . . 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 465 . . . 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 232 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )  e.  ( EE `  N ) )
14133adant1 1006 . 2  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( i  e.  ( 1 ... N
)  |->  ( ( A `
 i )  -  ( B `  i ) ) )  e.  ( EE `  N ) )
152, 14eqeltrd 2542 1  |-  ( (
ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201   RRcr 9393   1c1 9395    - cmin 9707   NNcn 10434   ...cfz 11555   EEcee 23287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-sub 9709  df-neg 9710  df-ee 23290
This theorem is referenced by: (None)
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