Theorem hashxp 11245

Description: The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

Assertion

Ref	Expression
hashxp	|- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )

Proof of Theorem hashxp

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4783	. . . 4 |- ( x = (/) -> ( x X. B ) = ( (/) X. B ) )
2	1	fveq2d 5694	. . 3 |- ( x = (/) -> (♯ ` ( x X. B ) ) = (♯ ` ( (/) X. B ) ) )
3		fveq2 5690	. . . 4 |- ( x = (/) -> (♯ ` x ) = (♯ ` (/) ) )
4	3	oveq1d 6090	. . 3 |- ( x = (/) -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) )
5	2, 4	eqeq12d 2253	. 2 |- ( x = (/) -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( (/) X. B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) ) )
6		xpeq1 4783	. . . 4 |- ( x = y -> ( x X. B ) = ( y X. B ) )
7	6	fveq2d 5694	. . 3 |- ( x = y -> (♯ ` ( x X. B ) ) = (♯ ` ( y X. B ) ) )
8		fveq2 5690	. . . 4 |- ( x = y -> (♯ ` x ) = (♯ ` y ) )
9	8	oveq1d 6090	. . 3 |- ( x = y -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) )
10	7, 9	eqeq12d 2253	. 2 |- ( x = y -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) )
11		xpeq1 4783	. . . 4 |- ( x = ( y u. { z } ) -> ( x X. B ) = ( ( y u. { z } ) X. B ) )
12	11	fveq2d 5694	. . 3 |- ( x = ( y u. { z } ) -> (♯ ` ( x X. B ) ) = (♯ ` ( ( y u. { z } ) X. B ) ) )
13		fveq2 5690	. . . 4 |- ( x = ( y u. { z } ) -> (♯ ` x ) = (♯ ` ( y u. { z } ) ) )
14	13	oveq1d 6090	. . 3 |- ( x = ( y u. { z } ) -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) )
15	12, 14	eqeq12d 2253	. 2 |- ( x = ( y u. { z } ) -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) ) )
16		xpeq1 4783	. . . 4 |- ( x = A -> ( x X. B ) = ( A X. B ) )
17	16	fveq2d 5694	. . 3 |- ( x = A -> (♯ ` ( x X. B ) ) = (♯ ` ( A X. B ) ) )
18		fveq2 5690	. . . 4 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
19	18	oveq1d 6090	. . 3 |- ( x = A -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )
20	17, 19	eqeq12d 2253	. 2 |- ( x = A -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) ) )
21		0xp 4850	. . . . 5 |- ( (/) X. B ) = (/)
22	21	fveq2i 5693	. . . 4 |- (♯ ` ( (/) X. B ) ) = (♯ ` (/) )
23		hash0 11213	. . . 4 |- (♯ ` (/) ) = 0
24	22, 23	eqtri 2259	. . 3 |- (♯ ` ( (/) X. B ) ) = 0
25	23	oveq1i 6085	. . . 4 |- ( (♯ ` (/) ) x. (♯ ` B ) ) = ( 0 x. (♯ ` B ) )
26		hashcl 11198	. . . . . . 7 |- ( B e. Fin -> (♯ ` B ) e. NN0 )
27	26	nn0cnd 9601	. . . . . 6 |- ( B e. Fin -> (♯ ` B ) e. CC )
28	27	mul02d 8709	. . . . 5 |- ( B e. Fin -> ( 0 x. (♯ ` B ) ) = 0 )
29	28	adantl 277	. . . 4 |- ( ( A e. Fin /\ B e. Fin ) -> ( 0 x. (♯ ` B ) ) = 0 )
30	25, 29	eqtrid 2283	. . 3 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` (/) ) x. (♯ ` B ) ) = 0 )
31	24, 30	eqtr4id 2290	. 2 |- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( (/) X. B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) )
32		oveq1 6082	. . . . 5 |- ( (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) -> ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) = ( ( (♯ ` y ) x. (♯ ` B ) ) + (♯ ` B ) ) )
33	32	adantl 277	. . . 4 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) -> ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) = ( ( (♯ ` y ) x. (♯ ` B ) ) + (♯ ` B ) ) )
34		xpundir 4827	. . . . . . 7 |- ( ( y u. { z } ) X. B ) = ( ( y X. B ) u. ( { z } X. B ) )
35	34	fveq2i 5693	. . . . . 6 |- (♯ ` ( ( y u. { z } ) X. B ) ) = (♯ ` ( ( y X. B ) u. ( { z } X. B ) ) )
36		simplr 533	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> y e. Fin )
37		simpllr 540	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> B e. Fin )
38		xpfi 7229	. . . . . . . . 9 |- ( ( y e. Fin /\ B e. Fin ) -> ( y X. B ) e. Fin )
39	36, 37, 38	syl2anc 415	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( y X. B ) e. Fin )
40		vex 2824	. . . . . . . . . . 11 |- z e. _V
41		snfig 7093	. . . . . . . . . . 11 |- ( z e. _V -> { z } e. Fin )
42	40, 41	ax-mp 5	. . . . . . . . . 10 |- { z } e. Fin
43		xpfi 7229	. . . . . . . . . 10 |- ( ( { z } e. Fin /\ B e. Fin ) -> ( { z } X. B ) e. Fin )
44	42, 43	mpan 428	. . . . . . . . 9 |- ( B e. Fin -> ( { z } X. B ) e. Fin )
45	44	ad3antlr 497	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) e. Fin )
46		simprr 537	. . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> z e. ( A \ y ) )
47	46	eldifbd 3232	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> -. z e. y )
48		inxp 4909	. . . . . . . . . 10 |- ( ( y X. B ) i^i ( { z } X. B ) ) = ( ( y i^i { z } ) X. ( B i^i B ) )
49		disjsn 3767	. . . . . . . . . . . . 13 |- ( ( y i^i { z } ) = (/) <-> -. z e. y )
50	49	biimpri 133	. . . . . . . . . . . 12 |- ( -. z e. y -> ( y i^i { z } ) = (/) )
51	50	xpeq1d 4792	. . . . . . . . . . 11 |- ( -. z e. y -> ( ( y i^i { z } ) X. ( B i^i B ) ) = ( (/) X. ( B i^i B ) ) )
52		0xp 4850	. . . . . . . . . . 11 |- ( (/) X. ( B i^i B ) ) = (/)
53	51, 52	eqtrdi 2287	. . . . . . . . . 10 |- ( -. z e. y -> ( ( y i^i { z } ) X. ( B i^i B ) ) = (/) )
54	48, 53	eqtrid 2283	. . . . . . . . 9 |- ( -. z e. y -> ( ( y X. B ) i^i ( { z } X. B ) ) = (/) )
55	47, 54	syl 14	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( ( y X. B ) i^i ( { z } X. B ) ) = (/) )
56		hashun 11223	. . . . . . . 8 |- ( ( ( y X. B ) e. Fin /\ ( { z } X. B ) e. Fin /\ ( ( y X. B ) i^i ( { z } X. B ) ) = (/) ) -> (♯ ` ( ( y X. B ) u. ( { z } X. B ) ) ) = ( (♯ ` ( y X. B ) ) + (♯ ` ( { z } X. B ) ) ) )
57	39, 45, 55, 56	syl3anc 1278	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` ( ( y X. B ) u. ( { z } X. B ) ) ) = ( (♯ ` ( y X. B ) ) + (♯ ` ( { z } X. B ) ) ) )
58	40	snex 4317	. . . . . . . . . . . 12 |- { z } e. _V
59	58	a1i 9	. . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> { z } e. _V )
60		xpcomeng 7116	. . . . . . . . . . 11 |- ( ( { z } e. _V /\ B e. Fin ) -> ( { z } X. B ) ~~ ( B X. { z } ) )
61	59, 37, 60	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) ~~ ( B X. { z } ) )
62	40	a1i 9	. . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> z e. _V )
63		xpsneng 7110	. . . . . . . . . . 11 |- ( ( B e. Fin /\ z e. _V ) -> ( B X. { z } ) ~~ B )
64	37, 62, 63	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( B X. { z } ) ~~ B )
65		entr 7061	. . . . . . . . . 10 |- ( ( ( { z } X. B ) ~~ ( B X. { z } ) /\ ( B X. { z } ) ~~ B ) -> ( { z } X. B ) ~~ B )
66	61, 64, 65	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) ~~ B )
67		hashen 11201	. . . . . . . . . 10 |- ( ( ( { z } X. B ) e. Fin /\ B e. Fin ) -> ( (♯ ` ( { z } X. B ) ) = (♯ ` B ) <-> ( { z } X. B ) ~~ B ) )
68	45, 37, 67	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( (♯ ` ( { z } X. B ) ) = (♯ ` B ) <-> ( { z } X. B ) ~~ B ) )
69	66, 68	mpbird 167	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` ( { z } X. B ) ) = (♯ ` B ) )
70	69	oveq2d 6091	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( (♯ ` ( y X. B ) ) + (♯ ` ( { z } X. B ) ) ) = ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) )
71	57, 70	eqtrd 2271	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` ( ( y X. B ) u. ( { z } X. B ) ) ) = ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) )
72	35, 71	eqtrid 2283	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) )
73	72	adantr 276	. . . 4 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) -> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) )
74		hashunsng 11226	. . . . . . . . 9 |- ( z e. _V -> ( ( y e. Fin /\ -. z e. y ) -> (♯ ` ( y u. { z } ) ) = ( (♯ ` y ) + 1 ) ) )
75	40, 74	ax-mp 5	. . . . . . . 8 |- ( ( y e. Fin /\ -. z e. y ) -> (♯ ` ( y u. { z } ) ) = ( (♯ ` y ) + 1 ) )
76	75	oveq1d 6090	. . . . . . 7 |- ( ( y e. Fin /\ -. z e. y ) -> ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) = ( ( (♯ ` y ) + 1 ) x. (♯ ` B ) ) )
77	36, 47, 76	syl2anc 415	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) = ( ( (♯ ` y ) + 1 ) x. (♯ ` B ) ) )
78		hashcl 11198	. . . . . . . . 9 |- ( y e. Fin -> (♯ ` y ) e. NN0 )
79	78	nn0cnd 9601	. . . . . . . 8 |- ( y e. Fin -> (♯ ` y ) e. CC )
80	36, 79	syl 14	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` y ) e. CC )
81	37, 27	syl 14	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` B ) e. CC )
82	80, 81	adddirp1d 8342	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( ( (♯ ` y ) + 1 ) x. (♯ ` B ) ) = ( ( (♯ ` y ) x. (♯ ` B ) ) + (♯ ` B ) ) )
83	77, 82	eqtrd 2271	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) = ( ( (♯ ` y ) x. (♯ ` B ) ) + (♯ ` B ) ) )
84	83	adantr 276	. . . 4 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) -> ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) = ( ( (♯ ` y ) x. (♯ ` B ) ) + (♯ ` B ) ) )
85	33, 73, 84	3eqtr4d 2281	. . 3 |- ( ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) -> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) )
86	85	ex 115	. 2 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) -> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) ) )
87		simpl 109	. 2 |- ( ( A e. Fin /\ B e. Fin ) -> A e. Fin )
88	5, 10, 15, 20, 31, 86, 87	findcard2sd 7186	1 |- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3705   class class class wbr 4125   X. cxp 4767   ` cfv 5372  (class class class)co 6075   ~~ cen 7010   Fincfn 7012   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174  ♯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-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  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-fz 10391  df-ihash 11193

This theorem is referenced by:  hashmap  11246  crth  12980  phimullem  12981  lgsquadlem3  16112