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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 6666 | . 2 ⊢ (♯‘𝑂) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
3 | fzfi 13328 | . . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin | |
4 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
5 | 4 | nnzi 11994 | . . 3 ⊢ 𝑀 ∈ ℤ |
6 | hashbc 13799 | . . 3 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})) | |
7 | 3, 5, 6 | mp2an 688 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
8 | ballotth.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
9 | 4, 8 | pm3.2i 471 | . . . . 5 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) |
10 | nnaddcl 11648 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
11 | nnnn0 11892 | . . . . 5 ⊢ ((𝑀 + 𝑁) ∈ ℕ → (𝑀 + 𝑁) ∈ ℕ0) | |
12 | 9, 10, 11 | mp2b 10 | . . . 4 ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
13 | hashfz1 13694 | . . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℕ0 → (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁) |
15 | 14 | oveq1i 7155 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = ((𝑀 + 𝑁)C𝑀) |
16 | 2, 7, 15 | 3eqtr2i 2847 | 1 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 𝒫 cpw 4535 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 1c1 10526 + caddc 10528 ℕcn 11626 ℕ0cn0 11885 ℤcz 11969 ...cfz 12880 Ccbc 13650 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-fac 13622 df-bc 13651 df-hash 13679 |
This theorem is referenced by: ballotlem2 31645 ballotth 31694 |
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