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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 10229 | . . 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 6904 | . 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 9201 | . . . 4 β’ ((π΄ β Fin β§ π΅ β Fin) β (π΄ βͺ π΅) β Fin) | |
4 | eqid 2727 | . . . . 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 14330 | . . . 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 1129 | . 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 9990 | . . . . 5 β’ (π΄ β Fin β (cardβπ΄) β Ο) | |
9 | ficardom 9990 | . . . . 5 β’ (π΅ β Fin β (cardβπ΅) β Ο) | |
10 | 4 | hashgadd 14374 | . . . . 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 594 | . . . 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 14330 | . . . . 5 β’ (π΄ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΄)) = (β―βπ΄)) |
13 | 4 | hashgval 14330 | . . . . 5 β’ (π΅ β Fin β ((rec((π₯ β V β¦ (π₯ + 1)), 0) βΎ Ο)β(cardβπ΅)) = (β―βπ΅)) |
14 | 12, 13 | oveqan12d 7443 | . . . 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 2767 | . . 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 1129 | . 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 2775 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3471 βͺ cun 3945 β© cin 3946 β c0 4324 β¦ cmpt 5233 βΎ cres 5682 βcfv 6551 (class class class)co 7424 Οcom 7874 reccrdg 8434 +o coa 8488 Fincfn 8968 cardccrd 9964 0cc0 11144 1c1 11145 + caddc 11147 β―chash 14327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-oadd 8495 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-dju 9930 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-hash 14328 |
This theorem is referenced by: hashun2 14380 hashun3 14381 hashunx 14383 hashunsng 14389 hashssdif 14409 hashxplem 14430 hashfun 14434 hashbclem 14449 hashf1lem2 14455 climcndslem1 15833 climcndslem2 15834 phiprmpw 16750 prmreclem5 16894 4sqlem11 16929 ppidif 27113 mumul 27131 ppiub 27155 lgsquadlem2 27332 lgsquadlem3 27333 numedglnl 28975 cusgrsizeinds 29284 eupth2eucrct 30045 numclwwlk3lem2 30212 ex-hash 30281 ballotlemgun 34149 ballotth 34162 subfacp1lem1 34794 subfacp1lem6 34800 poimirlem27 37125 sticksstones22 41644 eldioph2lem1 42183 |
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