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Mathbox for Thierry Arnoux |
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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 6905 | . 2 ⊢ (♯‘𝑂) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
3 | fzfi 13977 | . . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin | |
4 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
5 | 4 | nnzi 12624 | . . 3 ⊢ 𝑀 ∈ ℤ |
6 | hashbc 14452 | . . 3 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})) | |
7 | 3, 5, 6 | mp2an 690 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) |
8 | ballotth.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
9 | 4, 8 | pm3.2i 469 | . . . . 5 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) |
10 | nnaddcl 12273 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ) | |
11 | nnnn0 12517 | . . . . 5 ⊢ ((𝑀 + 𝑁) ∈ ℕ → (𝑀 + 𝑁) ∈ ℕ0) | |
12 | 9, 10, 11 | mp2b 10 | . . . 4 ⊢ (𝑀 + 𝑁) ∈ ℕ0 |
13 | hashfz1 14345 | . . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℕ0 → (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (♯‘(1...(𝑀 + 𝑁))) = (𝑀 + 𝑁) |
15 | 14 | oveq1i 7436 | . 2 ⊢ ((♯‘(1...(𝑀 + 𝑁)))C𝑀) = ((𝑀 + 𝑁)C𝑀) |
16 | 2, 7, 15 | 3eqtr2i 2762 | 1 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3430 𝒫 cpw 4606 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 1c1 11147 + caddc 11149 ℕcn 12250 ℕ0cn0 12510 ℤcz 12596 ...cfz 13524 Ccbc 14301 ♯chash 14329 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-seq 14007 df-fac 14273 df-bc 14302 df-hash 14330 |
This theorem is referenced by: ballotlem2 34141 ballotth 34190 |
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