Theorem hashun 11223

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.

Step	Hyp	Ref	Expression
1		isfi 7037	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 120	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	3ad2ant1 1049	. 2 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> E. n e. om A ~~ n )
4		isfi 7037	. . . . . 6 |- ( B e. Fin <-> E. m e. om B ~~ m )
5	4	biimpi 120	. . . . 5 |- ( B e. Fin -> E. m e. om B ~~ m )
6	5	3ad2ant2 1050	. . . 4 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> E. m e. om B ~~ m )
7	6	adantr 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 541	. . . . . 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 535	. . . . . 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 2238	. . . . . . 7 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11	10	omgadd 11220	. . . . . 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 ) ) )
12	8, 9, 11	syl2anc 415	. . . . 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 6743	. . . . . . 7 |- ( ( n e. om /\ m e. om ) -> ( n +o m ) e. om )
14	8, 9, 13	syl2anc 415	. . . . . 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 7040	. . . . . . 7 |- ( ( n +o m ) e. om -> ( n +o m ) ~~ ( n +o m ) )
16	14, 15	syl 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 11197	. . . . . 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 ) ) )
18	14, 16, 17	syl2anc 415	. . . . 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 2824	. . . . . . . 8 |- n e. _V
20	19	enref 7041	. . . . . . 7 |- n ~~ n
21		hashennn 11197	. . . . . . 7 |- ( ( n e. om /\ n ~~ n ) -> (♯ ` n ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
22	8, 20, 21	sylancl 417	. . . . . 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 2824	. . . . . . . 8 |- m e. _V
24	23	enref 7041	. . . . . . 7 |- m ~~ m
25		hashennn 11197	. . . . . . 7 |- ( ( m e. om /\ m ~~ m ) -> (♯ ` m ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
26	9, 24, 25	sylancl 417	. . . . . 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 ) )
27	22, 26	oveq12d 6093	. . . . 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 ) ) )
28	12, 18, 27	3eqtr4d 2281	. . . 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 1067	. . . . . 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 1068	. . . . . 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 1069	. . . . . 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 542	. . . . . 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 537	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~~ m )
34	29, 30, 31, 8, 9, 32, 33	hashunlem 11222	. . . . 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 7219	. . . . . . 7 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> ( A u. B ) e. Fin )
36	35	ad2antrr 492	. . . . . 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 7164	. . . . . . . 8 |- ( ( n +o m ) e. om -> ( n +o m ) e. Fin )
38	13, 37	syl 14	. . . . . . 7 |- ( ( n e. om /\ m e. om ) -> ( n +o m ) e. Fin )
39	8, 9, 38	syl2anc 415	. . . . . 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 11201	. . . . . 6 |- ( ( ( A u. B ) e. Fin /\ ( n +o m ) e. Fin ) -> ( (♯ ` ( A u. B ) ) = (♯ ` ( n +o m ) ) <-> ( A u. B ) ~~ ( n +o m ) ) )
41	36, 39, 40	syl2anc 415	. . . . 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 ) ) )
42	34, 41	mpbird 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 7164	. . . . . . . 8 |- ( n e. om -> n e. Fin )
44	8, 43	syl 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 11201	. . . . . . 7 |- ( ( A e. Fin /\ n e. Fin ) -> ( (♯ ` A ) = (♯ ` n ) <-> A ~~ n ) )
46	29, 44, 45	syl2anc 415	. . . . . 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 ) )
47	32, 46	mpbird 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 7164	. . . . . . . 8 |- ( m e. om -> m e. Fin )
49	9, 48	syl 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 11201	. . . . . . 7 |- ( ( B e. Fin /\ m e. Fin ) -> ( (♯ ` B ) = (♯ ` m ) <-> B ~~ m ) )
51	30, 49, 50	syl2anc 415	. . . . . 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 ) )
52	33, 51	mpbird 167	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> (♯ ` B ) = (♯ ` m ) )
53	47, 52	oveq12d 6093	. . . 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 ) ) )
54	28, 42, 53	3eqtr4d 2281	. . 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 ) ) )
55	7, 54	rexlimddv 2673	. 2 |- ( ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) /\ ( n e. om /\ A ~~ n ) ) -> (♯ ` ( A u. B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
56	3, 55	rexlimddv 2673	1 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> (♯ ` ( A u. B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   u. cun 3218   i^i cin 3219   (/)c0 3520   class class class wbr 4125   |-> cmpt 4187   omcom 4732   ` cfv 5372  (class class class)co 6075  freccfrec 6651   +o coa 6674   ~~ cen 7010   Fincfn 7012   0cc0 8169   1c1 8170   + caddc 8172   ZZcz 9623  ♯chash 11192

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-ihash 11193

This theorem is referenced by:  hashunsng  11226  fihashssdif  11237  hashxp  11245  hashfibclem  11260  hashf1lem2  11264  hashtpgim  11275  fsumconst  12199  phiprmpw  12978  4sqlem11  13158  ballotfilemgun  13246  ballotfilemth  13259  lgsquadlem2  16111  lgsquadlem3  16112  vtxdfifiun  16452