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Theorem eleesub 23304
Description: Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)
Hypothesis
Ref Expression
eleesub.1  |-  C  =  ( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )
Assertion
Ref Expression
eleesub  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
Distinct variable groups:    i, N    A, i    B, i
Allowed substitution hint:    C( i)

Proof of Theorem eleesub
StepHypRef Expression
1 eleesub.1 . 2  |-  C  =  ( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )
2 fveere 23294 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
3 fveere 23294 . . . . . 6  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
4 resubcl 9779 . . . . . 6  |-  ( ( ( A `  i
)  e.  RR  /\  ( B `  i )  e.  RR )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
52, 3, 4syl2an 477 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  i  e.  ( 1 ... N ) )  /\  ( B  e.  ( EE `  N )  /\  i  e.  ( 1 ... N
) ) )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
65anandirs 827 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
76ralrimiva 2827 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR )
8 eleenn 23289 . . . . 5  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
9 mptelee 23288 . . . . 5  |-  ( 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 ) )
108, 9syl 16 . . . 4  |-  ( 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 ) )
1110adantr 465 . . 3  |-  ( ( 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 ) )
127, 11mpbird 232 . 2  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( i  e.  ( 1 ... N ) 
|->  ( ( A `  i )  -  ( B `  i )
) )  e.  ( EE `  N ) )
131, 12syl5eqel 2544 1  |-  ( ( 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    = wceq 1370    e. wcel 1758   A.wral 2796    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   RRcr 9387   1c1 9389    - cmin 9701   NNcn 10428   ...cfz 11549   EEcee 23281
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-ltxr 9529  df-sub 9703  df-neg 9704  df-ee 23284
This theorem is referenced by: (None)
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