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Theorem isconc2 26007
Description: Concatenation with the empty set. (Contributed by FL, 14-Jan-2014.)
Assertion
Ref Expression
isconc2  |-  ( A  e.  B  ->  ( A  conc  (/) )  =  A )

Proof of Theorem isconc2
Dummy variables  x  y  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
2 0ex 4150 . . . 4  |-  (/)  e.  _V
32a1i 10 . . 3  |-  ( A  e.  B  ->  (/)  e.  _V )
42a1i 10 . . . 4  |-  ( ( A  e.  B  /\  A  =  (/) )  ->  (/) 
e.  _V )
51adantr 451 . . . . . 6  |-  ( ( A  e.  B  /\  (/)  =  (/) )  ->  A  e.  _V )
6 ovex 5883 . . . . . . . 8  |-  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  e.  _V
7 mptexg 5745 . . . . . . . 8  |-  ( ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  e.  _V  ->  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) )  e.  _V )
86, 7mp1i 11 . . . . . . 7  |-  ( ( A  e.  B  /\  -.  (/)  =  (/) )  -> 
( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) )  e.  _V )
9 unexg 4521 . . . . . . 7  |-  ( ( A  e.  B  /\  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) )  e.  _V )  ->  ( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) )  e.  _V )
108, 9syldan 456 . . . . . 6  |-  ( ( A  e.  B  /\  -.  (/)  =  (/) )  -> 
( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) )  e.  _V )
115, 10ifclda 3592 . . . . 5  |-  ( A  e.  B  ->  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) )  e. 
_V )
1211adantr 451 . . . 4  |-  ( ( A  e.  B  /\  -.  A  =  (/) )  ->  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) )  e. 
_V )
134, 12ifclda 3592 . . 3  |-  ( A  e.  B  ->  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  e.  _V )
14 eqeq1 2289 . . . . 5  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
15 id 19 . . . . . 6  |-  ( x  =  A  ->  x  =  A )
16 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  A  ->  ( # `
 x )  =  ( # `  A
) )
1716oveq1d 5873 . . . . . . . . 9  |-  ( x  =  A  ->  (
( # `  x )  +  1 )  =  ( ( # `  A
)  +  1 ) )
1816oveq1d 5873 . . . . . . . . 9  |-  ( x  =  A  ->  (
( # `  x )  +  ( # `  y
) )  =  ( ( # `  A
)  +  ( # `  y ) ) )
1917, 18oveq12d 5876 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( # `  x
)  +  1 ) ... ( ( # `  x )  +  (
# `  y )
) )  =  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  y )
) ) )
2016oveq2d 5874 . . . . . . . . 9  |-  ( x  =  A  ->  (
a  -  ( # `  x ) )  =  ( a  -  ( # `
 A ) ) )
2120fveq2d 5529 . . . . . . . 8  |-  ( x  =  A  ->  (
y `  ( a  -  ( # `  x
) ) )  =  ( y `  (
a  -  ( # `  A ) ) ) )
2219, 21mpteq12dv 4098 . . . . . . 7  |-  ( x  =  A  ->  (
a  e.  ( ( ( # `  x
)  +  1 ) ... ( ( # `  x )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  x ) ) ) )  =  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  A ) ) ) ) )
2315, 22uneq12d 3330 . . . . . 6  |-  ( x  =  A  ->  (
x  u.  ( a  e.  ( ( (
# `  x )  +  1 ) ... ( ( # `  x
)  +  ( # `  y ) ) ) 
|->  ( y `  (
a  -  ( # `  x ) ) ) ) )  =  ( A  u.  ( a  e.  ( ( (
# `  A )  +  1 ) ... ( ( # `  A
)  +  ( # `  y ) ) ) 
|->  ( y `  (
a  -  ( # `  A ) ) ) ) ) )
2415, 23ifeq12d 3581 . . . . 5  |-  ( x  =  A  ->  if ( y  =  (/) ,  x ,  ( x  u.  ( a  e.  ( ( ( # `  x )  +  1 ) ... ( (
# `  x )  +  ( # `  y
) ) )  |->  ( y `  ( a  -  ( # `  x
) ) ) ) ) )  =  if ( y  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  y
) ) )  |->  ( y `  ( a  -  ( # `  A
) ) ) ) ) ) )
2514, 24ifbieq2d 3585 . . . 4  |-  ( x  =  A  ->  if ( x  =  (/) ,  y ,  if ( y  =  (/) ,  x ,  ( x  u.  (
a  e.  ( ( ( # `  x
)  +  1 ) ... ( ( # `  x )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  x ) ) ) ) ) ) )  =  if ( A  =  (/) ,  y ,  if ( y  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  A ) ) ) ) ) ) ) )
26 id 19 . . . . 5  |-  ( y  =  (/)  ->  y  =  (/) )
27 eqeq1 2289 . . . . . 6  |-  ( y  =  (/)  ->  ( y  =  (/)  <->  (/)  =  (/) ) )
28 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( # `  y )  =  (
# `  (/) ) )
2928oveq2d 5874 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( (
# `  A )  +  ( # `  y
) )  =  ( ( # `  A
)  +  ( # `  (/) ) ) )
3029oveq2d 5874 . . . . . . . 8  |-  ( y  =  (/)  ->  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  y )
) )  =  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) ) )
31 fveq1 5524 . . . . . . . 8  |-  ( y  =  (/)  ->  ( y `
 ( a  -  ( # `  A ) ) )  =  (
(/) `  ( a  -  ( # `  A
) ) ) )
3230, 31mpteq12dv 4098 . . . . . . 7  |-  ( y  =  (/)  ->  ( a  e.  ( ( (
# `  A )  +  1 ) ... ( ( # `  A
)  +  ( # `  y ) ) ) 
|->  ( y `  (
a  -  ( # `  A ) ) ) )  =  ( a  e.  ( ( (
# `  A )  +  1 ) ... ( ( # `  A
)  +  ( # `  (/) ) ) ) 
|->  ( (/) `  ( a  -  ( # `  A
) ) ) ) )
3332uneq2d 3329 . . . . . 6  |-  ( y  =  (/)  ->  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  y
) ) )  |->  ( y `  ( a  -  ( # `  A
) ) ) ) )  =  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) )
3427, 33ifbieq2d 3585 . . . . 5  |-  ( y  =  (/)  ->  if ( y  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  A ) ) ) ) ) )  =  if (
(/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )
3526, 34ifeq12d 3581 . . . 4  |-  ( y  =  (/)  ->  if ( A  =  (/) ,  y ,  if ( y  =  (/) ,  A , 
( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  A ) ) ) ) ) ) )  =  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A , 
( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) ) )
36 df-conc 26005 . . . 4  |-  conc  =  ( x  e.  _V ,  y  e.  _V  |->  if ( x  =  (/) ,  y ,  if ( y  =  (/) ,  x ,  ( x  u.  ( a  e.  ( ( ( # `  x
)  +  1 ) ... ( ( # `  x )  +  (
# `  y )
) )  |->  ( y `
 ( a  -  ( # `  x ) ) ) ) ) ) ) )
3725, 35, 36ovmpt2g 5982 . . 3  |-  ( ( A  e.  _V  /\  (/) 
e.  _V  /\  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  e.  _V )  -> 
( A  conc  (/) )  =  if ( A  =  (/) ,  (/) ,  if (
(/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) ) )
381, 3, 13, 37syl3anc 1182 . 2  |-  ( A  e.  B  ->  ( A  conc  (/) )  =  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A , 
( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) ) )
39 iftrue 3571 . . . 4  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  =  (/) )
40 eqcom 2285 . . . . 5  |-  ( A  =  (/)  <->  (/)  =  A )
4140biimpi 186 . . . 4  |-  ( A  =  (/)  ->  (/)  =  A )
4239, 41eqtrd 2315 . . 3  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  =  A )
43 iffalse 3572 . . . 4  |-  ( -.  A  =  (/)  ->  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  =  if ( (/)  =  (/) ,  A , 
( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )
44 eqid 2283 . . . . 5  |-  (/)  =  (/)
45 iftrue 3571 . . . . 5  |-  ( (/)  =  (/)  ->  if ( (/)  =  (/) ,  A , 
( A  u.  (
a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) )  =  A )
4644, 45ax-mp 8 . . . 4  |-  if (
(/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A
)  +  1 ) ... ( ( # `  A )  +  (
# `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) )  =  A
4743, 46syl6eq 2331 . . 3  |-  ( -.  A  =  (/)  ->  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  =  A )
4842, 47pm2.61i 156 . 2  |-  if ( A  =  (/) ,  (/) ,  if ( (/)  =  (/) ,  A ,  ( A  u.  ( a  e.  ( ( ( # `  A )  +  1 ) ... ( (
# `  A )  +  ( # `  (/) ) ) )  |->  ( (/) `  (
a  -  ( # `  A ) ) ) ) ) ) )  =  A
4938, 48syl6eq 2331 1  |-  ( A  e.  B  ->  ( A  conc  (/) )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   (/)c0 3455   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    - cmin 9037   ...cfz 10782   #chash 11337    conc cconc 26004
This theorem is referenced by:  clscnc  26010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-conc 26005
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