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| Mirrors > Home > ILE Home > Th. List > hashxp | Unicode version | ||
| Description: The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) |
| Ref | Expression |
|---|---|
| hashxp |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xpeq1 4688 |
. . . 4
| |
| 2 | 1 | fveq2d 5579 |
. . 3
|
| 3 | fveq2 5575 |
. . . 4
| |
| 4 | 3 | oveq1d 5958 |
. . 3
|
| 5 | 2, 4 | eqeq12d 2219 |
. 2
|
| 6 | xpeq1 4688 |
. . . 4
| |
| 7 | 6 | fveq2d 5579 |
. . 3
|
| 8 | fveq2 5575 |
. . . 4
| |
| 9 | 8 | oveq1d 5958 |
. . 3
|
| 10 | 7, 9 | eqeq12d 2219 |
. 2
|
| 11 | xpeq1 4688 |
. . . 4
| |
| 12 | 11 | fveq2d 5579 |
. . 3
|
| 13 | fveq2 5575 |
. . . 4
| |
| 14 | 13 | oveq1d 5958 |
. . 3
|
| 15 | 12, 14 | eqeq12d 2219 |
. 2
|
| 16 | xpeq1 4688 |
. . . 4
| |
| 17 | 16 | fveq2d 5579 |
. . 3
|
| 18 | fveq2 5575 |
. . . 4
| |
| 19 | 18 | oveq1d 5958 |
. . 3
|
| 20 | 17, 19 | eqeq12d 2219 |
. 2
|
| 21 | 0xp 4754 |
. . . . 5
| |
| 22 | 21 | fveq2i 5578 |
. . . 4
|
| 23 | hash0 10939 |
. . . 4
| |
| 24 | 22, 23 | eqtri 2225 |
. . 3
|
| 25 | 23 | oveq1i 5953 |
. . . 4
|
| 26 | hashcl 10924 |
. . . . . . 7
| |
| 27 | 26 | nn0cnd 9349 |
. . . . . 6
|
| 28 | 27 | mul02d 8463 |
. . . . 5
|
| 29 | 28 | adantl 277 |
. . . 4
|
| 30 | 25, 29 | eqtrid 2249 |
. . 3
|
| 31 | 24, 30 | eqtr4id 2256 |
. 2
|
| 32 | oveq1 5950 |
. . . . 5
| |
| 33 | 32 | adantl 277 |
. . . 4
|
| 34 | xpundir 4731 |
. . . . . . 7
| |
| 35 | 34 | fveq2i 5578 |
. . . . . 6
|
| 36 | simplr 528 |
. . . . . . . . 9
| |
| 37 | simpllr 534 |
. . . . . . . . 9
| |
| 38 | xpfi 7028 |
. . . . . . . . 9
| |
| 39 | 36, 37, 38 | syl2anc 411 |
. . . . . . . 8
|
| 40 | vex 2774 |
. . . . . . . . . . 11
| |
| 41 | snfig 6905 |
. . . . . . . . . . 11
| |
| 42 | 40, 41 | ax-mp 5 |
. . . . . . . . . 10
|
| 43 | xpfi 7028 |
. . . . . . . . . 10
| |
| 44 | 42, 43 | mpan 424 |
. . . . . . . . 9
|
| 45 | 44 | ad3antlr 493 |
. . . . . . . 8
|
| 46 | simprr 531 |
. . . . . . . . . 10
| |
| 47 | 46 | eldifbd 3177 |
. . . . . . . . 9
|
| 48 | inxp 4811 |
. . . . . . . . . 10
| |
| 49 | disjsn 3694 |
. . . . . . . . . . . . 13
| |
| 50 | 49 | biimpri 133 |
. . . . . . . . . . . 12
|
| 51 | 50 | xpeq1d 4697 |
. . . . . . . . . . 11
|
| 52 | 0xp 4754 |
. . . . . . . . . . 11
| |
| 53 | 51, 52 | eqtrdi 2253 |
. . . . . . . . . 10
|
| 54 | 48, 53 | eqtrid 2249 |
. . . . . . . . 9
|
| 55 | 47, 54 | syl 14 |
. . . . . . . 8
|
| 56 | hashun 10948 |
. . . . . . . 8
| |
| 57 | 39, 45, 55, 56 | syl3anc 1249 |
. . . . . . 7
|
| 58 | 40 | snex 4228 |
. . . . . . . . . . . 12
|
| 59 | 58 | a1i 9 |
. . . . . . . . . . 11
|
| 60 | xpcomeng 6922 |
. . . . . . . . . . 11
| |
| 61 | 59, 37, 60 | syl2anc 411 |
. . . . . . . . . 10
|
| 62 | 40 | a1i 9 |
. . . . . . . . . . 11
|
| 63 | xpsneng 6916 |
. . . . . . . . . . 11
| |
| 64 | 37, 62, 63 | syl2anc 411 |
. . . . . . . . . 10
|
| 65 | entr 6875 |
. . . . . . . . . 10
| |
| 66 | 61, 64, 65 | syl2anc 411 |
. . . . . . . . 9
|
| 67 | hashen 10927 |
. . . . . . . . . 10
| |
| 68 | 45, 37, 67 | syl2anc 411 |
. . . . . . . . 9
|
| 69 | 66, 68 | mpbird 167 |
. . . . . . . 8
|
| 70 | 69 | oveq2d 5959 |
. . . . . . 7
|
| 71 | 57, 70 | eqtrd 2237 |
. . . . . 6
|
| 72 | 35, 71 | eqtrid 2249 |
. . . . 5
|
| 73 | 72 | adantr 276 |
. . . 4
|
| 74 | hashunsng 10950 |
. . . . . . . . 9
| |
| 75 | 40, 74 | ax-mp 5 |
. . . . . . . 8
|
| 76 | 75 | oveq1d 5958 |
. . . . . . 7
|
| 77 | 36, 47, 76 | syl2anc 411 |
. . . . . 6
|
| 78 | hashcl 10924 |
. . . . . . . . 9
| |
| 79 | 78 | nn0cnd 9349 |
. . . . . . . 8
|
| 80 | 36, 79 | syl 14 |
. . . . . . 7
|
| 81 | 37, 27 | syl 14 |
. . . . . . 7
|
| 82 | 80, 81 | adddirp1d 8098 |
. . . . . 6
|
| 83 | 77, 82 | eqtrd 2237 |
. . . . 5
|
| 84 | 83 | adantr 276 |
. . . 4
|
| 85 | 33, 73, 84 | 3eqtr4d 2247 |
. . 3
|
| 86 | 85 | ex 115 |
. 2
|
| 87 | simpl 109 |
. 2
| |
| 88 | 5, 10, 15, 20, 31, 86, 87 | findcard2sd 6988 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-irdg 6455 df-frec 6476 df-1o 6501 df-oadd 6505 df-er 6619 df-en 6827 df-dom 6828 df-fin 6829 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-inn 9036 df-n0 9295 df-z 9372 df-uz 9648 df-fz 10130 df-ihash 10919 |
| This theorem is referenced by: crth 12517 phimullem 12518 lgsquadlem3 15527 |
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