Theorem hashxp 11216

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 4768	. . . 4 |- ( x = (/) -> ( x X. B ) = ( (/) X. B ) )
2	1	fveq2d 5679	. . 3 |- ( x = (/) -> (♯ ` ( x X. B ) ) = (♯ ` ( (/) X. B ) ) )
3		fveq2 5675	. . . 4 |- ( x = (/) -> (♯ ` x ) = (♯ ` (/) ) )
4	3	oveq1d 6073	. . 3 |- ( x = (/) -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) )
5	2, 4	eqeq12d 2249	. 2 |- ( x = (/) -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( (/) X. B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) ) )
6		xpeq1 4768	. . . 4 |- ( x = y -> ( x X. B ) = ( y X. B ) )
7	6	fveq2d 5679	. . 3 |- ( x = y -> (♯ ` ( x X. B ) ) = (♯ ` ( y X. B ) ) )
8		fveq2 5675	. . . 4 |- ( x = y -> (♯ ` x ) = (♯ ` y ) )
9	8	oveq1d 6073	. . 3 |- ( x = y -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) )
10	7, 9	eqeq12d 2249	. 2 |- ( x = y -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) )
11		xpeq1 4768	. . . 4 |- ( x = ( y u. { z } ) -> ( x X. B ) = ( ( y u. { z } ) X. B ) )
12	11	fveq2d 5679	. . 3 |- ( x = ( y u. { z } ) -> (♯ ` ( x X. B ) ) = (♯ ` ( ( y u. { z } ) X. B ) ) )
13		fveq2 5675	. . . 4 |- ( x = ( y u. { z } ) -> (♯ ` x ) = (♯ ` ( y u. { z } ) ) )
14	13	oveq1d 6073	. . 3 |- ( x = ( y u. { z } ) -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) )
15	12, 14	eqeq12d 2249	. 2 |- ( x = ( y u. { z } ) -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) ) )
16		xpeq1 4768	. . . 4 |- ( x = A -> ( x X. B ) = ( A X. B ) )
17	16	fveq2d 5679	. . 3 |- ( x = A -> (♯ ` ( x X. B ) ) = (♯ ` ( A X. B ) ) )
18		fveq2 5675	. . . 4 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
19	18	oveq1d 6073	. . 3 |- ( x = A -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )
20	17, 19	eqeq12d 2249	. 2 |- ( x = A -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) ) )
21		0xp 4835	. . . . 5 |- ( (/) X. B ) = (/)
22	21	fveq2i 5678	. . . 4 |- (♯ ` ( (/) X. B ) ) = (♯ ` (/) )
23		hash0 11184	. . . 4 |- (♯ ` (/) ) = 0
24	22, 23	eqtri 2255	. . 3 |- (♯ ` ( (/) X. B ) ) = 0
25	23	oveq1i 6068	. . . 4 |- ( (♯ ` (/) ) x. (♯ ` B ) ) = ( 0 x. (♯ ` B ) )
26		hashcl 11169	. . . . . . 7 |- ( B e. Fin -> (♯ ` B ) e. NN0 )
27	26	nn0cnd 9572	. . . . . 6 |- ( B e. Fin -> (♯ ` B ) e. CC )
28	27	mul02d 8682	. . . . 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 2279	. . 3 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` (/) ) x. (♯ ` B ) ) = 0 )
31	24, 30	eqtr4id 2286	. 2 |- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( (/) X. B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) )
32		oveq1 6065	. . . . 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 4812	. . . . . . 7 |- ( ( y u. { z } ) X. B ) = ( ( y X. B ) u. ( { z } X. B ) )
35	34	fveq2i 5678	. . . . . 6 |- (♯ ` ( ( y u. { z } ) X. B ) ) = (♯ ` ( ( y X. B ) u. ( { z } X. B ) ) )
36		simplr 529	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> y e. Fin )
37		simpllr 536	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> B e. Fin )
38		xpfi 7205	. . . . . . . . 9 |- ( ( y e. Fin /\ B e. Fin ) -> ( y X. B ) e. Fin )
39	36, 37, 38	syl2anc 411	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( y X. B ) e. Fin )
40		vex 2818	. . . . . . . . . . 11 |- z e. _V
41		snfig 7069	. . . . . . . . . . 11 |- ( z e. _V -> { z } e. Fin )
42	40, 41	ax-mp 5	. . . . . . . . . 10 |- { z } e. Fin
43		xpfi 7205	. . . . . . . . . 10 |- ( ( { z } e. Fin /\ B e. Fin ) -> ( { z } X. B ) e. Fin )
44	42, 43	mpan 424	. . . . . . . . 9 |- ( B e. Fin -> ( { z } X. B ) e. Fin )
45	44	ad3antlr 493	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) e. Fin )
46		simprr 533	. . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> z e. ( A \ y ) )
47	46	eldifbd 3226	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> -. z e. y )
48		inxp 4894	. . . . . . . . . 10 |- ( ( y X. B ) i^i ( { z } X. B ) ) = ( ( y i^i { z } ) X. ( B i^i B ) )
49		disjsn 3756	. . . . . . . . . . . . 13 |- ( ( y i^i { z } ) = (/) <-> -. z e. y )
50	49	biimpri 133	. . . . . . . . . . . 12 |- ( -. z e. y -> ( y i^i { z } ) = (/) )
51	50	xpeq1d 4777	. . . . . . . . . . 11 |- ( -. z e. y -> ( ( y i^i { z } ) X. ( B i^i B ) ) = ( (/) X. ( B i^i B ) ) )
52		0xp 4835	. . . . . . . . . . 11 |- ( (/) X. ( B i^i B ) ) = (/)
53	51, 52	eqtrdi 2283	. . . . . . . . . 10 |- ( -. z e. y -> ( ( y i^i { z } ) X. ( B i^i B ) ) = (/) )
54	48, 53	eqtrid 2279	. . . . . . . . 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 11194	. . . . . . . 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 1274	. . . . . . 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 4303	. . . . . . . . . . . 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 7092	. . . . . . . . . . 11 |- ( ( { z } e. _V /\ B e. Fin ) -> ( { z } X. B ) ~~ ( B X. { z } ) )
61	59, 37, 60	syl2anc 411	. . . . . . . . . 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 7086	. . . . . . . . . . 11 |- ( ( B e. Fin /\ z e. _V ) -> ( B X. { z } ) ~~ B )
64	37, 62, 63	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( B X. { z } ) ~~ B )
65		entr 7037	. . . . . . . . . 10 |- ( ( ( { z } X. B ) ~~ ( B X. { z } ) /\ ( B X. { z } ) ~~ B ) -> ( { z } X. B ) ~~ B )
66	61, 64, 65	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) ~~ B )
67		hashen 11172	. . . . . . . . . 10 |- ( ( ( { z } X. B ) e. Fin /\ B e. Fin ) -> ( (♯ ` ( { z } X. B ) ) = (♯ ` B ) <-> ( { z } X. B ) ~~ B ) )
68	45, 37, 67	syl2anc 411	. . . . . . . . 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 6074	. . . . . . 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 2267	. . . . . 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 2279	. . . . 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 11197	. . . . . . . . 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 6073	. . . . . . 7 |- ( ( y e. Fin /\ -. z e. y ) -> ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) = ( ( (♯ ` y ) + 1 ) x. (♯ ` B ) ) )
77	36, 47, 76	syl2anc 411	. . . . . 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 11169	. . . . . . . . 9 |- ( y e. Fin -> (♯ ` y ) e. NN0 )
79	78	nn0cnd 9572	. . . . . . . 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 8316	. . . . . 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 2267	. . . . 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 2277	. . 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 7162	1 |- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   \ cdif 3211   u. cun 3212   i^i cin 3213   C_ wss 3214   (/)c0 3512   {csn 3694   class class class wbr 4114   X. cxp 4752   ` cfv 5357  (class class class)co 6058   ~~ cen 6986   Fincfn 6988   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148  ♯chash 11163

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-ihash 11164

This theorem is referenced by:  hashmap  11217  crth  12946  phimullem  12947  lgsquadlem3  16078