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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 | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ficardun 9779 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +o (card‘𝐵))) | |
2 | 1 | fveq2d 6699 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(𝐴 ∪ 𝐵))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +o (card‘𝐵)))) |
3 | unfi 8828 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
4 | eqid 2736 | . . . . 5 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
5 | 4 | hashgval 13864 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(𝐴 ∪ 𝐵))) = (♯‘(𝐴 ∪ 𝐵))) |
6 | 3, 5 | syl 17 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(𝐴 ∪ 𝐵))) = (♯‘(𝐴 ∪ 𝐵))) |
7 | 6 | 3adant3 1134 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(𝐴 ∪ 𝐵))) = (♯‘(𝐴 ∪ 𝐵))) |
8 | ficardom 9542 | . . . . 5 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) | |
9 | ficardom 9542 | . . . . 5 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω) | |
10 | 4 | hashgadd 13909 | . . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +o (card‘𝐵))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))) |
11 | 8, 9, 10 | syl2an 599 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +o (card‘𝐵))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))) |
12 | 4 | hashgval 13864 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴)) |
13 | 4 | hashgval 13864 | . . . . 5 ⊢ (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵)) |
14 | 12, 13 | oveqan12d 7210 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = ((♯‘𝐴) + (♯‘𝐵))) |
15 | 11, 14 | eqtrd 2771 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +o (card‘𝐵))) = ((♯‘𝐴) + (♯‘𝐵))) |
16 | 15 | 3adant3 1134 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +o (card‘𝐵))) = ((♯‘𝐴) + (♯‘𝐵))) |
17 | 2, 7, 16 | 3eqtr3d 2779 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 ∩ cin 3852 ∅c0 4223 ↦ cmpt 5120 ↾ cres 5538 ‘cfv 6358 (class class class)co 7191 ωcom 7622 reccrdg 8123 +o coa 8177 Fincfn 8604 cardccrd 9516 0cc0 10694 1c1 10695 + caddc 10697 ♯chash 13861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-hash 13862 |
This theorem is referenced by: hashun2 13915 hashun3 13916 hashunx 13918 hashunsng 13924 hashssdif 13944 hashxplem 13965 hashfun 13969 hashbclem 13981 hashf1lem2 13987 climcndslem1 15376 climcndslem2 15377 phiprmpw 16292 prmreclem5 16436 4sqlem11 16471 ppidif 25999 mumul 26017 ppiub 26039 lgsquadlem2 26216 lgsquadlem3 26217 numedglnl 27189 cusgrsizeinds 27494 eupth2eucrct 28254 numclwwlk3lem2 28421 ex-hash 28490 ballotlemgun 32157 ballotth 32170 subfacp1lem1 32808 subfacp1lem6 32814 poimirlem27 35490 eldioph2lem1 40226 |
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