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Theorem hashun 11661
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 8087 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( card `  ( A  u.  B
) )  =  ( ( card `  A
)  +o  ( card `  B ) ) )
21fveq2d 5735 . 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 7377 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B
)  e.  Fin )
4 eqid 2438 . . . . 5  |-  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  0 )  |`  om )  =  ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om )
54hashgval 11626 . . . 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 978 . 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 7853 . . . . 5  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
9 ficardom 7853 . . . . 5  |-  ( B  e.  Fin  ->  ( card `  B )  e. 
om )
104hashgadd 11656 . . . . 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 465 . . . 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 11626 . . . . 5  |-  ( A  e.  Fin  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  A ) )  =  ( # `  A
) )
134hashgval 11626 . . . . 5  |-  ( B  e.  Fin  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om ) `  ( card `  B ) )  =  ( # `  B
) )
1412, 13oveqan12d 6103 . . . 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 2470 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  0 )  |`  om ) `  ( ( card `  A
)  +o  ( card `  B ) ) )  =  ( ( # `  A )  +  (
# `  B )
) )
16153adant3 978 . 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 2478 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    i^i cin 3321   (/)c0 3630    e. cmpt 4269   omcom 4848    |` cres 4883   ` cfv 5457  (class class class)co 6084   reccrdg 6670    +o coa 6724   Fincfn 7112   cardccrd 7827   0cc0 8995   1c1 8996    + caddc 8998   #chash 11623
This theorem is referenced by:  hashun2  11662  hashun3  11663  hashunx  11665  hashunsng  11670  hashssdif  11682  hashxplem  11701  hashfun  11705  hashbclem  11706  hashf1lem2  11710  climcndslem1  12634  climcndslem2  12635  phiprmpw  13170  prmreclem5  13293  4sqlem11  13328  ppidif  20951  mumul  20969  ppiub  20993  lgsquadlem2  21144  lgsquadlem3  21145  cusgrasizeinds  21490  vdgrfiun  21678  ballotlemgun  24787  ballotth  24800  subfacp1lem1  24870  subfacp1lem6  24876  eldioph2lem1  26832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-hash 11624
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