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Mirrors > Home > MPE Home > Th. List > hashun | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
hashun | β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (β―β(π΄ βͺ π΅)) = ((β―βπ΄) + (β―βπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ficardun 10143 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ(π΄ βͺ π΅)) = ((cardβπ΄) +o (cardβπ΅))) | |
2 | 1 | fveq2d 6851 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(π΄ βͺ π΅))) = ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β((cardβπ΄) +o (cardβπ΅)))) |
3 | unfi 9123 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin) β (π΄ βͺ π΅) β Fin) | |
4 | eqid 2737 | . . . . 5 β’ (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) = (rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο) | |
5 | 4 | hashgval 14240 | . . . 4 β’ ((π΄ βͺ π΅) β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(π΄ βͺ π΅))) = (β―β(π΄ βͺ π΅))) |
6 | 3, 5 | syl 17 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(π΄ βͺ π΅))) = (β―β(π΄ βͺ π΅))) |
7 | 6 | 3adant3 1133 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβ(π΄ βͺ π΅))) = (β―β(π΄ βͺ π΅))) |
8 | ficardom 9904 | . . . . 5 β’ (π΄ β Fin β (cardβπ΄) β Ο) | |
9 | ficardom 9904 | . . . . 5 β’ (π΅ β Fin β (cardβπ΅) β Ο) | |
10 | 4 | hashgadd 14284 | . . . . 5 β’ (((cardβπ΄) β Ο β§ (cardβπ΅) β Ο) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β((cardβπ΄) +o (cardβπ΅))) = (((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) + ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)))) |
11 | 8, 9, 10 | syl2an 597 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β((cardβπ΄) +o (cardβπ΅))) = (((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) + ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)))) |
12 | 4 | hashgval 14240 | . . . . 5 β’ (π΄ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = (β―βπ΄)) |
13 | 4 | hashgval 14240 | . . . . 5 β’ (π΅ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)) = (β―βπ΅)) |
14 | 12, 13 | oveqan12d 7381 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin) β (((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) + ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅))) = ((β―βπ΄) + (β―βπ΅))) |
15 | 11, 14 | eqtrd 2777 | . . 3 β’ ((π΄ β Fin β§ π΅ β Fin) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β((cardβπ΄) +o (cardβπ΅))) = ((β―βπ΄) + (β―βπ΅))) |
16 | 15 | 3adant3 1133 | . 2 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β((cardβπ΄) +o (cardβπ΅))) = ((β―βπ΄) + (β―βπ΅))) |
17 | 2, 7, 16 | 3eqtr3d 2785 | 1 β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (β―β(π΄ βͺ π΅)) = ((β―βπ΄) + (β―βπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3448 βͺ cun 3913 β© cin 3914 β c0 4287 β¦ cmpt 5193 βΎ cres 5640 βcfv 6501 (class class class)co 7362 Οcom 7807 reccrdg 8360 +o coa 8414 Fincfn 8890 cardccrd 9878 0cc0 11058 1c1 11059 + caddc 11061 β―chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-hash 14238 |
This theorem is referenced by: hashun2 14290 hashun3 14291 hashunx 14293 hashunsng 14299 hashssdif 14319 hashxplem 14340 hashfun 14344 hashbclem 14356 hashf1lem2 14362 climcndslem1 15741 climcndslem2 15742 phiprmpw 16655 prmreclem5 16799 4sqlem11 16834 ppidif 26528 mumul 26546 ppiub 26568 lgsquadlem2 26745 lgsquadlem3 26746 numedglnl 28137 cusgrsizeinds 28442 eupth2eucrct 29203 numclwwlk3lem2 29370 ex-hash 29439 ballotlemgun 33164 ballotth 33177 subfacp1lem1 33813 subfacp1lem6 33819 poimirlem27 36134 sticksstones22 40605 eldioph2lem1 41112 |
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