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Theorem hashun 10113
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 6432 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 118 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
323ad2ant1 962 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  E. n  e.  om  A  ~~  n
)
4 isfi 6432 . . . . . 6  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 118 . . . . 5  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
653ad2ant2 963 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  E. m  e.  om  B  ~~  m
)
76adantr 270 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  (
n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m )
8 simplrl 502 . . . . . 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 498 . . . . . 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 2085 . . . . . . 7  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )
1110omgadd 10110 . . . . . 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 403 . . . . 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 6197 . . . . . . 7  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  +o  m
)  e.  om )
148, 9, 13syl2anc 403 . . . . . 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 6435 . . . . . . 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 10088 . . . . . 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 403 . . . . 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 2618 . . . . . . . 8  |-  n  e. 
_V
2019enref 6436 . . . . . . 7  |-  n  ~~  n
21 hashennn 10088 . . . . . . 7  |-  ( ( n  e.  om  /\  n  ~~  n )  -> 
( `  n )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
228, 20, 21sylancl 404 . . . . . 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 2618 . . . . . . . 8  |-  m  e. 
_V
2423enref 6436 . . . . . . 7  |-  m  ~~  m
25 hashennn 10088 . . . . . . 7  |-  ( ( m  e.  om  /\  m  ~~  m )  -> 
( `  m )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
269, 24, 25sylancl 404 . . . . . 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 5633 . . . . 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 2127 . . . 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 980 . . . . . 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 981 . . . . . 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 982 . . . . . 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 503 . . . . . 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 499 . . . . . 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 10112 . . . . 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 6586 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  e. 
Fin )
3635ad2antrr 472 . . . . . 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 6542 . . . . . . . 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 403 . . . . . 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 10092 . . . . . 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 403 . . . . 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 165 . . . 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 6542 . . . . . . . 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 10092 . . . . . . 7  |-  ( ( A  e.  Fin  /\  n  e.  Fin )  ->  ( ( `  A
)  =  ( `  n
)  <->  A  ~~  n ) )
4629, 44, 45syl2anc 403 . . . . . 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 165 . . . . 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 6542 . . . . . . . 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 10092 . . . . . . 7  |-  ( ( B  e.  Fin  /\  m  e.  Fin )  ->  ( ( `  B
)  =  ( `  m
)  <->  B  ~~  m ) )
5130, 49, 50syl2anc 403 . . . . . 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 165 . . . . 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 5633 . . . 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 2127 . . 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 2489 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  /\  (
n  e.  om  /\  A  ~~  n ) )  ->  ( `  ( A  u.  B ) )  =  ( ( `  A
)  +  ( `  B
) ) )
563, 55rexlimddv 2489 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 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436   E.wrex 2356    u. cun 2986    i^i cin 2987   (/)c0 3275   class class class wbr 3822    |-> cmpt 3876   omcom 4380   ` cfv 4983  (class class class)co 5615  freccfrec 6111    +o coa 6134    ~~ cen 6409   Fincfn 6411   0cc0 7297   1c1 7298    + caddc 7300   ZZcz 8686  ♯chash 10083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-addcom 7392  ax-addass 7394  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-0id 7400  ax-rnegex 7401  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-ltadd 7408
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-ilim 4172  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-frec 6112  df-1o 6137  df-oadd 6141  df-er 6246  df-en 6412  df-dom 6413  df-fin 6414  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-inn 8361  df-n0 8610  df-z 8687  df-uz 8955  df-ihash 10084
This theorem is referenced by:  hashunsng  10115  fihashssdif  10126  hashxp  10134  phiprmpw  11104
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