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Theorem hashxp 10572
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.
StepHypRef Expression
1 xpeq1 4553 . . . 4 |- ( x = (/) -> ( x X. B ) = ( (/) X. B ) )
21fveq2d 5425 . . 3 |- ( x = (/) -> (♯ ` ( x X. B ) ) = (♯ ` ( (/) X. B ) ) )
3 fveq2 5421 . . . 4 |- ( x = (/) -> (♯ ` x ) = (♯ ` (/) ) )
43oveq1d 5789 . . 3 |- ( x = (/) -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) )
52, 4eqeq12d 2154 . 2 |- ( x = (/) -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( (/) X. B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) ) )
6 xpeq1 4553 . . . 4 |- ( x = y -> ( x X. B ) = ( y X. B ) )
76fveq2d 5425 . . 3 |- ( x = y -> (♯ ` ( x X. B ) ) = (♯ ` ( y X. B ) ) )
8 fveq2 5421 . . . 4 |- ( x = y -> (♯ ` x ) = (♯ ` y ) )
98oveq1d 5789 . . 3 |- ( x = y -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) )
107, 9eqeq12d 2154 . 2 |- ( x = y -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) ) )
11 xpeq1 4553 . . . 4 |- ( x = ( y u. { z } ) -> ( x X. B ) = ( ( y u. { z } ) X. B ) )
1211fveq2d 5425 . . 3 |- ( x = ( y u. { z } ) -> (♯ ` ( x X. B ) ) = (♯ ` ( ( y u. { z } ) X. B ) ) )
13 fveq2 5421 . . . 4 |- ( x = ( y u. { z } ) -> (♯ ` x ) = (♯ ` ( y u. { z } ) ) )
1413oveq1d 5789 . . 3 |- ( x = ( y u. { z } ) -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) )
1512, 14eqeq12d 2154 . 2 |- ( x = ( y u. { z } ) -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( ( y u. { z } ) X. B ) ) = ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) ) )
16 xpeq1 4553 . . . 4 |- ( x = A -> ( x X. B ) = ( A X. B ) )
1716fveq2d 5425 . . 3 |- ( x = A -> (♯ ` ( x X. B ) ) = (♯ ` ( A X. B ) ) )
18 fveq2 5421 . . . 4 |- ( x = A -> (♯ ` x ) = (♯ ` A ) )
1918oveq1d 5789 . . 3 |- ( x = A -> ( (♯ ` x ) x. (♯ ` B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )
2017, 19eqeq12d 2154 . 2 |- ( x = A -> ( (♯ ` ( x X. B ) ) = ( (♯ ` x ) x. (♯ ` B ) ) <-> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) ) )
21 hash0 10543 . . . . 5 |- (♯ ` (/) ) = 0
2221oveq1i 5784 . . . 4 |- ( (♯ ` (/) ) x. (♯ ` B ) ) = ( 0 x. (♯ ` B ) )
23 hashcl 10527 . . . . . . 7 |- ( B e. Fin -> (♯ ` B ) e. NN0 )
2423nn0cnd 9032 . . . . . 6 |- ( B e. Fin -> (♯ ` B ) e. CC )
2524mul02d 8154 . . . . 5 |- ( B e. Fin -> ( 0 x. (♯ ` B ) ) = 0 )
2625adantl 275 . . . 4 |- ( ( A e. Fin /\ B e. Fin ) -> ( 0 x. (♯ ` B ) ) = 0 )
2722, 26syl5eq 2184 . . 3 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` (/) ) x. (♯ ` B ) ) = 0 )
28 0xp 4619 . . . . 5 |- ( (/) X. B ) = (/)
2928fveq2i 5424 . . . 4 |- (♯ ` ( (/) X. B ) ) = (♯ ` (/) )
3029, 21eqtri 2160 . . 3 |- (♯ ` ( (/) X. B ) ) = 0
3127, 30syl6reqr 2191 . 2 |- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( (/) X. B ) ) = ( (♯ ` (/) ) x. (♯ ` B ) ) )
32 oveq1 5781 . . . . 5 |- ( (♯ ` ( y X. B ) ) = ( (♯ ` y ) x. (♯ ` B ) ) -> ( (♯ ` ( y X. B ) ) + (♯ ` B ) ) = ( ( (♯ ` y ) x. (♯ ` B ) ) + (♯ ` B ) ) )
3332adantl 275 . . . 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 4596 . . . . . . 7 |- ( ( y u. { z } ) X. B ) = ( ( y X. B ) u. ( { z } X. B ) )
3534fveq2i 5424 . . . . . 6 |- (♯ ` ( ( y u. { z } ) X. B ) ) = (♯ ` ( ( y X. B ) u. ( { z } X. B ) ) )
36 simplr 519 . . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> y e. Fin )
37 simpllr 523 . . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> B e. Fin )
38 xpfi 6818 . . . . . . . . 9 |- ( ( y e. Fin /\ B e. Fin ) -> ( y X. B ) e. Fin )
3936, 37, 38syl2anc 408 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( y X. B ) e. Fin )
40 vex 2689 . . . . . . . . . . 11 |- z e. _V
41 snfig 6708 . . . . . . . . . . 11 |- ( z e. _V -> { z } e. Fin )
4240, 41ax-mp 5 . . . . . . . . . 10 |- { z } e. Fin
43 xpfi 6818 . . . . . . . . . 10 |- ( ( { z } e. Fin /\ B e. Fin ) -> ( { z } X. B ) e. Fin )
4442, 43mpan 420 . . . . . . . . 9 |- ( B e. Fin -> ( { z } X. B ) e. Fin )
4544ad3antlr 484 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) e. Fin )
46 simprr 521 . . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> z e. ( A \ y ) )
4746eldifbd 3083 . . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> -. z e. y )
48 inxp 4673 . . . . . . . . . 10 |- ( ( y X. B ) i^i ( { z } X. B ) ) = ( ( y i^i { z } ) X. ( B i^i B ) )
49 disjsn 3585 . . . . . . . . . . . . 13 |- ( ( y i^i { z } ) = (/) <-> -. z e. y )
5049biimpri 132 . . . . . . . . . . . 12 |- ( -. z e. y -> ( y i^i { z } ) = (/) )
5150xpeq1d 4562 . . . . . . . . . . 11 |- ( -. z e. y -> ( ( y i^i { z } ) X. ( B i^i B ) ) = ( (/) X. ( B i^i B ) ) )
52 0xp 4619 . . . . . . . . . . 11 |- ( (/) X. ( B i^i B ) ) = (/)
5351, 52syl6eq 2188 . . . . . . . . . 10 |- ( -. z e. y -> ( ( y i^i { z } ) X. ( B i^i B ) ) = (/) )
5448, 53syl5eq 2184 . . . . . . . . 9 |- ( -. z e. y -> ( ( y X. B ) i^i ( { z } X. B ) ) = (/) )
5547, 54syl 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 10551 . . . . . . . 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 ) ) ) )
5739, 45, 55, 56syl3anc 1216 . . . . . . 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 ) ) ) )
5840snex 4109 . . . . . . . . . . . 12 |- { z } e. _V
5958a1i 9 . . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> { z } e. _V )
60 xpcomeng 6722 . . . . . . . . . . 11 |- ( ( { z } e. _V /\ B e. Fin ) -> ( { z } X. B ) ~~ ( B X. { z } ) )
6159, 37, 60syl2anc 408 . . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) ~~ ( B X. { z } ) )
6240a1i 9 . . . . . . . . . . 11 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> z e. _V )
63 xpsneng 6716 . . . . . . . . . . 11 |- ( ( B e. Fin /\ z e. _V ) -> ( B X. { z } ) ~~ B )
6437, 62, 63syl2anc 408 . . . . . . . . . 10 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( B X. { z } ) ~~ B )
65 entr 6678 . . . . . . . . . 10 |- ( ( ( { z } X. B ) ~~ ( B X. { z } ) /\ ( B X. { z } ) ~~ B ) -> ( { z } X. B ) ~~ B )
6661, 64, 65syl2anc 408 . . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( { z } X. B ) ~~ B )
67 hashen 10530 . . . . . . . . . 10 |- ( ( ( { z } X. B ) e. Fin /\ B e. Fin ) -> ( (♯ ` ( { z } X. B ) ) = (♯ ` B ) <-> ( { z } X. B ) ~~ B ) )
6845, 37, 67syl2anc 408 . . . . . . . . 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 ) )
6966, 68mpbird 166 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` ( { z } X. B ) ) = (♯ ` B ) )
7069oveq2d 5790 . . . . . . 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 ) ) )
7157, 70eqtrd 2172 . . . . . 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 ) ) )
7235, 71syl5eq 2184 . . . . 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 ) ) )
7372adantr 274 . . . 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 10553 . . . . . . . . 9 |- ( z e. _V -> ( ( y e. Fin /\ -. z e. y ) -> (♯ ` ( y u. { z } ) ) = ( (♯ ` y ) + 1 ) ) )
7540, 74ax-mp 5 . . . . . . . 8 |- ( ( y e. Fin /\ -. z e. y ) -> (♯ ` ( y u. { z } ) ) = ( (♯ ` y ) + 1 ) )
7675oveq1d 5789 . . . . . . 7 |- ( ( y e. Fin /\ -. z e. y ) -> ( (♯ ` ( y u. { z } ) ) x. (♯ ` B ) ) = ( ( (♯ ` y ) + 1 ) x. (♯ ` B ) ) )
7736, 47, 76syl2anc 408 . . . . . 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 10527 . . . . . . . . 9 |- ( y e. Fin -> (♯ ` y ) e. NN0 )
7978nn0cnd 9032 . . . . . . . 8 |- ( y e. Fin -> (♯ ` y ) e. CC )
8036, 79syl 14 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` y ) e. CC )
8137, 24syl 14 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> (♯ ` B ) e. CC )
8280, 81adddirp1d 7792 . . . . . 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 ) ) )
8377, 82eqtrd 2172 . . . . 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 ) ) )
8483adantr 274 . . . 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 ) ) )
8533, 73, 843eqtr4d 2182 . . 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 ) ) )
8685ex 114 . 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 108 . 2 |- ( ( A e. Fin /\ B e. Fin ) -> A e. Fin )
885, 10, 15, 20, 31, 86, 87findcard2sd 6786 1 |- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   \ cdif 3068   u. cun 3069   i^i cin 3070   C_ wss 3071   (/)c0 3363   {csn 3527   class class class wbr 3929   X. cxp 4537   ` cfv 5123  (class class class)co 5774   ~~ cen 6632   Fincfn 6634   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625  ♯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-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  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-fz 9791  df-ihash 10522
This theorem is referenced by:  crth  11900  phimullem  11901
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