![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemfelz | Structured version Visualization version GIF version |
Description: (𝐹‘𝐶) has values in ℤ. (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} |
ballotth.p | ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) |
ballotth.f | ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) |
ballotlemfval.c | ⊢ (𝜑 → 𝐶 ∈ 𝑂) |
ballotlemfval.j | ⊢ (𝜑 → 𝐽 ∈ ℤ) |
Ref | Expression |
---|---|
ballotlemfelz | ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ballotth.m | . . 3 ⊢ 𝑀 ∈ ℕ | |
2 | ballotth.n | . . 3 ⊢ 𝑁 ∈ ℕ | |
3 | ballotth.o | . . 3 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
4 | ballotth.p | . . 3 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) | |
5 | ballotth.f | . . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) | |
6 | ballotlemfval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑂) | |
7 | ballotlemfval.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ ℤ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ballotlemfval 33129 | . 2 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶)))) |
9 | fzfi 13884 | . . . . . 6 ⊢ (1...𝐽) ∈ Fin | |
10 | inss1 4193 | . . . . . 6 ⊢ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽) | |
11 | ssfi 9124 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∩ 𝐶) ∈ Fin) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 ⊢ ((1...𝐽) ∩ 𝐶) ∈ Fin |
13 | hashcl 14263 | . . . . 5 ⊢ (((1...𝐽) ∩ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0 |
15 | 14 | nn0zi 12535 | . . 3 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ |
16 | difss 4096 | . . . . . 6 ⊢ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽) | |
17 | ssfi 9124 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∖ 𝐶) ∈ Fin) | |
18 | 9, 16, 17 | mp2an 691 | . . . . 5 ⊢ ((1...𝐽) ∖ 𝐶) ∈ Fin |
19 | hashcl 14263 | . . . . 5 ⊢ (((1...𝐽) ∖ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0 |
21 | 20 | nn0zi 12535 | . . 3 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ |
22 | zsubcl 12552 | . . 3 ⊢ (((♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ ∧ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ) | |
23 | 15, 21, 22 | mp2an 691 | . 2 ⊢ ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ |
24 | 8, 23 | eqeltrdi 2846 | 1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3410 ∖ cdif 3912 ∩ cin 3914 ⊆ wss 3915 𝒫 cpw 4565 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 Fincfn 8890 1c1 11059 + caddc 11061 − cmin 11392 / cdiv 11819 ℕcn 12160 ℕ0cn0 12420 ℤcz 12506 ...cfz 13431 ♯chash 14237 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: ballotlemfc0 33132 ballotlemfcc 33133 ballotlemodife 33137 ballotlemic 33146 ballotlem1c 33147 ballotlemfrceq 33168 ballotlemfrcn0 33169 |
Copyright terms: Public domain | W3C validator |