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Theorem hashun 10752
Description: The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
hashun  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( `  ( A  u.  B )
)  =  ( ( `  A )  +  ( `  B ) ) )

Proof of Theorem hashun
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6751 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 120 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
323ad2ant1 1018 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  E. n  e.  om  A  ~~  n
)
4 isfi 6751 . . . . . 6  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 120 . . . . 5  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
653ad2ant2 1019 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  E. m  e.  om  B  ~~  m
)
76adantr 276 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  (
n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m )
8 simplrl 535 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  e.  om )
9 simprl 529 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  e.  om )
10 eqid 2175 . . . . . . 7  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )
1110omgadd 10749 . . . . . 6  |-  ( ( n  e.  om  /\  m  e.  om )  ->  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  ( n  +o  m
) )  =  ( (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n )  +  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  m
) ) )
128, 9, 11syl2anc 411 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
(frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  ( n  +o  m
) )  =  ( (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n )  +  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  m
) ) )
13 nnacl 6471 . . . . . . 7  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  +o  m
)  e.  om )
148, 9, 13syl2anc 411 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( n  +o  m
)  e.  om )
15 enrefg 6754 . . . . . . 7  |-  ( ( n  +o  m )  e.  om  ->  (
n  +o  m ) 
~~  ( n  +o  m ) )
1614, 15syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( n  +o  m
)  ~~  ( n  +o  m ) )
17 hashennn 10726 . . . . . 6  |-  ( ( ( n  +o  m
)  e.  om  /\  ( n  +o  m
)  ~~  ( n  +o  m ) )  -> 
( `  ( n  +o  m ) )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  ( n  +o  m
) ) )
1814, 16, 17syl2anc 411 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  ( n  +o  m ) )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  ( n  +o  m
) ) )
19 vex 2738 . . . . . . . 8  |-  n  e. 
_V
2019enref 6755 . . . . . . 7  |-  n  ~~  n
21 hashennn 10726 . . . . . . 7  |-  ( ( n  e.  om  /\  n  ~~  n )  -> 
( `  n )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
228, 20, 21sylancl 413 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  n )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
23 vex 2738 . . . . . . . 8  |-  m  e. 
_V
2423enref 6755 . . . . . . 7  |-  m  ~~  m
25 hashennn 10726 . . . . . . 7  |-  ( ( m  e.  om  /\  m  ~~  m )  -> 
( `  m )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
269, 24, 25sylancl 413 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  m )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
2722, 26oveq12d 5883 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( ( `  n
)  +  ( `  m
) )  =  ( (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n )  +  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  m
) ) )
2812, 18, 273eqtr4d 2218 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  ( n  +o  m ) )  =  ( ( `  n
)  +  ( `  m
) ) )
29 simpll1 1036 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  e.  Fin )
30 simpll2 1037 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  B  e.  Fin )
31 simpll3 1038 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( A  i^i  B
)  =  (/) )
32 simplrr 536 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~~  n )
33 simprr 531 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  B  ~~  m )
3429, 30, 31, 8, 9, 32, 33hashunlem 10751 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( A  u.  B
)  ~~  ( n  +o  m ) )
35 unfidisj 6911 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  e. 
Fin )
3635ad2antrr 488 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( A  u.  B
)  e.  Fin )
37 nnfi 6862 . . . . . . . 8  |-  ( ( n  +o  m )  e.  om  ->  (
n  +o  m )  e.  Fin )
3813, 37syl 14 . . . . . . 7  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  +o  m
)  e.  Fin )
398, 9, 38syl2anc 411 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( n  +o  m
)  e.  Fin )
40 hashen 10730 . . . . . 6  |-  ( ( ( A  u.  B
)  e.  Fin  /\  ( n  +o  m
)  e.  Fin )  ->  ( ( `  ( A  u.  B )
)  =  ( `  (
n  +o  m ) )  <->  ( A  u.  B )  ~~  (
n  +o  m ) ) )
4136, 39, 40syl2anc 411 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( ( `  ( A  u.  B )
)  =  ( `  (
n  +o  m ) )  <->  ( A  u.  B )  ~~  (
n  +o  m ) ) )
4234, 41mpbird 167 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  ( A  u.  B ) )  =  ( `  ( n  +o  m ) ) )
43 nnfi 6862 . . . . . . . 8  |-  ( n  e.  om  ->  n  e.  Fin )
448, 43syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  e.  Fin )
45 hashen 10730 . . . . . . 7  |-  ( ( A  e.  Fin  /\  n  e.  Fin )  ->  ( ( `  A
)  =  ( `  n
)  <->  A  ~~  n ) )
4629, 44, 45syl2anc 411 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( ( `  A
)  =  ( `  n
)  <->  A  ~~  n ) )
4732, 46mpbird 167 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  A )  =  ( `  n )
)
48 nnfi 6862 . . . . . . . 8  |-  ( m  e.  om  ->  m  e.  Fin )
499, 48syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  e.  Fin )
50 hashen 10730 . . . . . . 7  |-  ( ( B  e.  Fin  /\  m  e.  Fin )  ->  ( ( `  B
)  =  ( `  m
)  <->  B  ~~  m ) )
5130, 49, 50syl2anc 411 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( ( `  B
)  =  ( `  m
)  <->  B  ~~  m ) )
5233, 51mpbird 167 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  B )  =  ( `  m )
)
5347, 52oveq12d 5883 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( ( `  A
)  +  ( `  B
) )  =  ( ( `  n )  +  ( `  m )
) )
5428, 42, 533eqtr4d 2218 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( `  ( A  u.  B ) )  =  ( ( `  A
)  +  ( `  B
) ) )
557, 54rexlimddv 2597 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  (
n  e.  om  /\  A  ~~  n ) )  ->  ( `  ( A  u.  B ) )  =  ( ( `  A
)  +  ( `  B
) ) )
563, 55rexlimddv 2597 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( `  ( A  u.  B )
)  =  ( ( `  A )  +  ( `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   E.wrex 2454    u. cun 3125    i^i cin 3126   (/)c0 3420   class class class wbr 3998    |-> cmpt 4059   omcom 4583   ` cfv 5208  (class class class)co 5865  freccfrec 6381    +o coa 6404    ~~ cen 6728   Fincfn 6730   0cc0 7786   1c1 7787    + caddc 7789   ZZcz 9224  ♯chash 10721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-oadd 6411  df-er 6525  df-en 6731  df-dom 6732  df-fin 6733  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-ihash 10722
This theorem is referenced by:  hashunsng  10754  fihashssdif  10765  hashxp  10773  fsumconst  11429  phiprmpw  12188
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