Theorem hashun 10876

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 6815	. . . 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 1020	. 2 |- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> E. n e. om A ~~ n )
4		isfi 6815	. . . . . 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 1021	. . . 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 2193	. . . . . . 7 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
11	10	omgadd 10873	. . . . . 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 6533	. . . . . . 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 6818	. . . . . . 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 10851	. . . . . 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 2763	. . . . . . . 8 |- n e. _V
20	19	enref 6819	. . . . . . 7 |- n ~~ n
21		hashennn 10851	. . . . . . 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 2763	. . . . . . . 8 |- m e. _V
24	23	enref 6819	. . . . . . 7 |- m ~~ m
25		hashennn 10851	. . . . . . 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 5936	. . . . 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 2236	. . . 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 1038	. . . . . 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 1039	. . . . . 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 1040	. . . . . 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 10875	. . . . 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 6978	. . . . . . 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 6928	. . . . . . . 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 10855	. . . . . 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 6928	. . . . . . . 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 10855	. . . . . . 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 6928	. . . . . . . 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 10855	. . . . . . 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 5936	. . . 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 2236	. . 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 2616	. 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 2616	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 980   = wceq 1364   e. wcel 2164   E.wrex 2473   u. cun 3151   i^i cin 3152   (/)c0 3446   class class class wbr 4029   |-> cmpt 4090   omcom 4622   ` cfv 5254  (class class class)co 5918  freccfrec 6443   +o coa 6466   ~~ cen 6792   Fincfn 6794   0cc0 7872   1c1 7873   + caddc 7875   ZZcz 9317  ♯chash 10846

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-ihash 10847

This theorem is referenced by:  hashunsng  10878  fihashssdif  10889  hashxp  10897  fsumconst  11597  phiprmpw  12360  4sqlem11  12539