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Theorem hashun 11611
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 variable  x is distinct from all other variables.
StepHypRef Expression
1 ficardun 8038 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( A  u.  B
) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
21fveq2d 5691 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  ( A  u.  B
) ) )  =  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om ) `  ( ( card `  A
)  +o  ( card `  B ) ) ) )
3 unfi 7333 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  Fin )
4 eqid 2404 . . . . 5  |-  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  =  ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )
54hashgval 11576 . . . 4  |-  ( ( A  u.  B )  e.  Fin  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  ( A  u.  B ) ) )  =  ( # `  ( A  u.  B )
) )
63, 5syl 16 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om ) `  ( card `  ( A  u.  B )
) )  =  (
# `  ( A  u.  B ) ) )
763adant3 977 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  ( A  u.  B
) ) )  =  ( # `  ( A  u.  B )
) )
8 ficardom 7804 . . . . 5  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
9 ficardom 7804 . . . . 5  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
104hashgadd 11606 . . . . 5  |-  ( ( ( card `  A
)  e.  om  /\  ( card `  B )  e.  om )  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  (
( card `  A )  +o  ( card `  B
) ) )  =  ( ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om ) `  ( card `  A
) )  +  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  B ) ) ) )
118, 9, 10syl2an 464 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om ) `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  A ) )  +  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om ) `  ( card `  B
) ) ) )
124hashgval 11576 . . . . 5  |-  ( A  e.  Fin  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  A ) )  =  ( # `  A
) )
134hashgval 11576 . . . . 5  |-  ( B  e.  Fin  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  B ) )  =  ( # `  B
) )
1412, 13oveqan12d 6059 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om ) `  ( card `  A
) )  +  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  B ) ) )  =  ( (
# `  A )  +  ( # `  B
) ) )
1511, 14eqtrd 2436 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om ) `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( ( # `  A )  +  (
# `  B )
) )
16153adant3 977 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( (
card `  A )  +o  ( card `  B
) ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
172, 7, 163eqtr3d 2444 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    i^i cin 3279   (/)c0 3588    e. cmpt 4226   omcom 4804    |` cres 4839   ` cfv 5413  (class class class)co 6040   reccrdg 6626    +o coa 6680   Fincfn 7068   cardccrd 7778   0cc0 8946   1c1 8947    + caddc 8949   #chash 11573
This theorem is referenced by:  hashun2  11612  hashun3  11613  hashunx  11615  hashunsng  11620  hashssdif  11632  hashxplem  11651  hashfun  11655  hashbclem  11656  hashf1lem2  11660  climcndslem1  12584  climcndslem2  12585  phiprmpw  13120  prmreclem5  13243  4sqlem11  13278  ppidif  20899  mumul  20917  ppiub  20941  lgsquadlem2  21092  lgsquadlem3  21093  cusgrasizeinds  21438  vdgrfiun  21626  ballotlemgun  24735  ballotth  24748  subfacp1lem1  24818  subfacp1lem6  24824  eldioph2lem1  26708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-hash 11574
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