Theorem hashun 10785

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 6761	. . . 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 1018	. 2 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> E. n e. om A ~~ n )
4		isfi 6761	. . . . . 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 1019	. . . 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 535	. . . . . 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 529	. . . . . 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 2177	. . . . . . 7 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11	10	omgadd 10782	. . . . . 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 411	. . . . 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 6481	. . . . . . 7 |- ( ( n e. om /\ m e. om ) -> ( n +o m ) e. om )
14	8, 9, 13	syl2anc 411	. . . . . 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 6764	. . . . . . 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 10760	. . . . . 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 411	. . . . 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 2741	. . . . . . . 8 |- n e. _V
20	19	enref 6765	. . . . . . 7 |- n ~~ n
21		hashennn 10760	. . . . . . 7 |- ( ( n e. om /\ n ~~ n ) -> (♯ ` n ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
22	8, 20, 21	sylancl 413	. . . . . 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 2741	. . . . . . . 8 |- m e. _V
24	23	enref 6765	. . . . . . 7 |- m ~~ m
25		hashennn 10760	. . . . . . 7 |- ( ( m e. om /\ m ~~ m ) -> (♯ ` m ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
26	9, 24, 25	sylancl 413	. . . . . 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 5893	. . . . 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 2220	. . . 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 1036	. . . . . 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 1037	. . . . . 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 1038	. . . . . 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 536	. . . . . 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 531	. . . . . 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 10784	. . . . 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 6921	. . . . . . 7 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> ( A u. B ) e. Fin )
36	35	ad2antrr 488	. . . . . 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 6872	. . . . . . . 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 411	. . . . . 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 10764	. . . . . 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 411	. . . . 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 6872	. . . . . . . 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 10764	. . . . . . 7 |- ( ( A e. Fin /\ n e. Fin ) -> ( (♯ ` A ) = (♯ ` n ) <-> A ~~ n ) )
46	29, 44, 45	syl2anc 411	. . . . . 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 6872	. . . . . . . 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 10764	. . . . . . 7 |- ( ( B e. Fin /\ m e. Fin ) -> ( (♯ ` B ) = (♯ ` m ) <-> B ~~ m ) )
51	30, 49, 50	syl2anc 411	. . . . . 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 5893	. . . 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 2220	. . 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 2599	. 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 2599	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 978   = wceq 1353   e. wcel 2148   E.wrex 2456   u. cun 3128   i^i cin 3129   (/)c0 3423   class class class wbr 4004   |-> cmpt 4065   omcom 4590   ` cfv 5217  (class class class)co 5875  freccfrec 6391   +o coa 6414   ~~ cen 6738   Fincfn 6740   0cc0 7811   1c1 7812   + caddc 7814   ZZcz 9253  ♯chash 10755

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-ihash 10756

This theorem is referenced by:  hashunsng  10787  fihashssdif  10798  hashxp  10806  fsumconst  11462  phiprmpw  12222