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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 6709 | . 2 ⊢ (♯‘𝑂) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
3 | fzfi 13528 | . . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin | |
4 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
5 | 4 | nnzi 12184 | . . 3 ⊢ 𝑀 ∈ ℤ |
6 | hashbc 14000 | . . 3 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})) | |
7 | 3, 5, 6 | mp2an 692 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
8 | ballotth.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
9 | 4, 8 | pm3.2i 474 | . . . . 5 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) |
10 | nnaddcl 11836 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
11 | nnnn0 12080 | . . . . 5 ⊢ ((𝑀 + 𝑁) ∈ ℕ → (𝑀 + 𝑁) ∈ ℕ0) | |
12 | 9, 10, 11 | mp2b 10 | . . . 4 ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
13 | hashfz1 13895 | . . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℕ0 → (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁) |
15 | 14 | oveq1i 7212 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = ((𝑀 + 𝑁)C𝑀) |
16 | 2, 7, 15 | 3eqtr2i 2768 | 1 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3058 𝒫 cpw 4503 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 1c1 10713 + caddc 10715 ℕcn 11813 ℕ0cn0 12073 ℤcz 12159 ...cfz 13078 Ccbc 13851 ♯chash 13879 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-seq 13558 df-fac 13823 df-bc 13852 df-hash 13880 |
This theorem is referenced by: ballotlem2 32139 ballotth 32188 |
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