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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlem1 | Structured version Visualization version GIF version |
Description: The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.) |
Ref | Expression |
---|---|
ballotth.m | ⊢ 𝑀 ∈ ℕ |
ballotth.n | ⊢ 𝑁 ∈ ℕ |
ballotth.o | ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} |
Ref | Expression |
---|---|
ballotlem1 | ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ballotth.o | . . 3 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
2 | 1 | fveq2i 6849 | . 2 ⊢ (♯‘𝑂) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
3 | fzfi 13886 | . . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin | |
4 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
5 | 4 | nnzi 12535 | . . 3 ⊢ 𝑀 ∈ ℤ |
6 | hashbc 14359 | . . 3 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})) | |
7 | 3, 5, 6 | mp2an 691 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
8 | ballotth.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
9 | 4, 8 | pm3.2i 472 | . . . . 5 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) |
10 | nnaddcl 12184 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
11 | nnnn0 12428 | . . . . 5 ⊢ ((𝑀 + 𝑁) ∈ ℕ → (𝑀 + 𝑁) ∈ ℕ0) | |
12 | 9, 10, 11 | mp2b 10 | . . . 4 ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
13 | hashfz1 14255 | . . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℕ0 → (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁) |
15 | 14 | oveq1i 7371 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = ((𝑀 + 𝑁)C𝑀) |
16 | 2, 7, 15 | 3eqtr2i 2767 | 1 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 𝒫 cpw 4564 ‘cfv 6500 (class class class)co 7361 Fincfn 8889 1c1 11060 + caddc 11062 ℕcn 12161 ℕ0cn0 12421 ℤcz 12507 ...cfz 13433 Ccbc 14211 ♯chash 14239 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-fz 13434 df-seq 13916 df-fac 14183 df-bc 14212 df-hash 14240 |
This theorem is referenced by: ballotlem2 33152 ballotth 33201 |
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