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Theorem eleesub 24416
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 24406 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
3 fveere 24406 . . . . . 6  |-  ( ( B  e.  ( EE
`  N )  /\  i  e.  ( 1 ... N ) )  ->  ( B `  i )  e.  RR )
4 resubcl 9874 . . . . . 6  |-  ( ( ( A `  i
)  e.  RR  /\  ( B `  i )  e.  RR )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
52, 3, 4syl2an 475 . . . . 5  |-  ( ( ( A  e.  ( EE `  N )  /\  i  e.  ( 1 ... N ) )  /\  ( B  e.  ( EE `  N )  /\  i  e.  ( 1 ... N
) ) )  -> 
( ( A `  i )  -  ( B `  i )
)  e.  RR )
65anandirs 829 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  i  e.  ( 1 ... N
) )  ->  (
( A `  i
)  -  ( B `
 i ) )  e.  RR )
76ralrimiva 2868 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  ->  A. i  e.  (
1 ... N ) ( ( A `  i
)  -  ( B `
 i ) )  e.  RR )
8 eleenn 24401 . . . . 5  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
9 mptelee 24400 . . . . 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 463 . . 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 2546 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 367    = wceq 1398    e. wcel 1823   A.wral 2804    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482    - cmin 9796   NNcn 10531   ...cfz 11675   EEcee 24393
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-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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
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-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  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-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-ee 24396
This theorem is referenced by: (None)
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