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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 31747 | . 2 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶)))) |
9 | fzfi 13341 | . . . . . 6 ⊢ (1...𝐽) ∈ Fin | |
10 | inss1 4205 | . . . . . 6 ⊢ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽) | |
11 | ssfi 8738 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∩ 𝐶) ∈ Fin) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ ((1...𝐽) ∩ 𝐶) ∈ Fin |
13 | hashcl 13718 | . . . . 5 ⊢ (((1...𝐽) ∩ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0 |
15 | 14 | nn0zi 12008 | . . 3 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ |
16 | difss 4108 | . . . . . 6 ⊢ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽) | |
17 | ssfi 8738 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∖ 𝐶) ∈ Fin) | |
18 | 9, 16, 17 | mp2an 690 | . . . . 5 ⊢ ((1...𝐽) ∖ 𝐶) ∈ Fin |
19 | hashcl 13718 | . . . . 5 ⊢ (((1...𝐽) ∖ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0 |
21 | 20 | nn0zi 12008 | . . 3 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ |
22 | zsubcl 12025 | . . 3 ⊢ (((♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ ∧ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ) | |
23 | 15, 21, 22 | mp2an 690 | . 2 ⊢ ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ |
24 | 8, 23 | eqeltrdi 2921 | 1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 1c1 10538 + caddc 10540 − cmin 10870 / cdiv 11297 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 ...cfz 12893 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: ballotlemfc0 31750 ballotlemfcc 31751 ballotlemodife 31755 ballotlemic 31764 ballotlem1c 31765 ballotlemfrceq 31786 ballotlemfrcn0 31787 |
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