Theorem hashun 10551

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 6655	. . . 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 6655	. . . . . 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 2139	. . . . . . 7 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11	10	omgadd 10548	. . . . . 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 6376	. . . . . . 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 6658	. . . . . . 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 10526	. . . . . 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 2689	. . . . . . . 8 |- n e. _V
20	19	enref 6659	. . . . . . 7 |- n ~~ n
21		hashennn 10526	. . . . . . 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 2689	. . . . . . . 8 |- m e. _V
24	23	enref 6659	. . . . . . 7 |- m ~~ m
25		hashennn 10526	. . . . . . 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 5792	. . . . 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 2182	. . . 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 10550	. . . . 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 6810	. . . . . . 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 6766	. . . . . . . 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 10530	. . . . . 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 6766	. . . . . . . 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 10530	. . . . . . 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 6766	. . . . . . . 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 10530	. . . . . . 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 5792	. . . 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 2182	. . 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 2554	. 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 2554	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 2417   u. cun 3069   i^i cin 3070   (/)c0 3363   class class class wbr 3929   |-> cmpt 3989   omcom 4504   ` cfv 5123  (class class class)co 5774  freccfrec 6287   +o coa 6310   ~~ cen 6632   Fincfn 6634   0cc0 7620   1c1 7621   + caddc 7623   ZZcz 9054  ♯chash 10521

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-ihash 10522

This theorem is referenced by:  hashunsng  10553  fihashssdif  10564  hashxp  10572  fsumconst  11223  phiprmpw  11898