Theorem hashun 10544

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 6648	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 119	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	3ad2ant1 1002	. 2 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> E. n e. om A ~~ n )
4		isfi 6648	. . . . . 6 |- ( B e. Fin <-> E. m e. om B ~~ m )
5	4	biimpi 119	. . . . 5 |- ( B e. Fin -> E. m e. om B ~~ m )
6	5	3ad2ant2 1003	. . . 4 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> E. m e. om B ~~ m )
7	6	adantr 274	. . 3 |- ( ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
8		simplrl 524	. . . . . 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 520	. . . . . 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 2137	. . . . . . 7 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11	10	omgadd 10541	. . . . . 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 408	. . . . 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 6369	. . . . . . 7 |- ( ( n e. om /\ m e. om ) -> ( n +o m ) e. om )
14	8, 9, 13	syl2anc 408	. . . . . 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 6651	. . . . . . 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 10519	. . . . . 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 408	. . . . 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 2684	. . . . . . . 8 |- n e. _V
20	19	enref 6652	. . . . . . 7 |- n ~~ n
21		hashennn 10519	. . . . . . 7 |- ( ( n e. om /\ n ~~ n ) -> (♯ ` n ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
22	8, 20, 21	sylancl 409	. . . . . 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 2684	. . . . . . . 8 |- m e. _V
24	23	enref 6652	. . . . . . 7 |- m ~~ m
25		hashennn 10519	. . . . . . 7 |- ( ( m e. om /\ m ~~ m ) -> (♯ ` m ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
26	9, 24, 25	sylancl 409	. . . . . 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 5785	. . . . 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 2180	. . . 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 1020	. . . . . 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 1021	. . . . . 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 1022	. . . . . 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 525	. . . . . 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 521	. . . . . 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 10543	. . . . 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 6803	. . . . . . 7 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> ( A u. B ) e. Fin )
36	35	ad2antrr 479	. . . . . 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 6759	. . . . . . . 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 408	. . . . . 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 10523	. . . . . 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 408	. . . . 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 166	. . . 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 6759	. . . . . . . 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 10523	. . . . . . 7 |- ( ( A e. Fin /\ n e. Fin ) -> ( (♯ ` A ) = (♯ ` n ) <-> A ~~ n ) )
46	29, 44, 45	syl2anc 408	. . . . . 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 166	. . . . 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 6759	. . . . . . . 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 10523	. . . . . . 7 |- ( ( B e. Fin /\ m e. Fin ) -> ( (♯ ` B ) = (♯ ` m ) <-> B ~~ m ) )
51	30, 49, 50	syl2anc 408	. . . . . 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 166	. . . . 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 5785	. . . 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 2180	. . 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 2552	. 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 2552	1 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> (♯ ` ( A u. B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   u. cun 3064   i^i cin 3065   (/)c0 3358   class class class wbr 3924   |-> cmpt 3984   omcom 4499   ` cfv 5118  (class class class)co 5767  freccfrec 6280   +o coa 6303   ~~ cen 6625   Fincfn 6627   0cc0 7613   1c1 7614   + caddc 7616   ZZcz 9047  ♯chash 10514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-ihash 10515

This theorem is referenced by:  hashunsng  10546  fihashssdif  10557  hashxp  10565  fsumconst  11216  phiprmpw  11887