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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 34492 | . 2 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶)))) | 
| 9 | fzfi 14013 | . . . . . 6 ⊢ (1...𝐽) ∈ Fin | |
| 10 | inss1 4237 | . . . . . 6 ⊢ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽) | |
| 11 | ssfi 9213 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∩ 𝐶) ∈ Fin) | |
| 12 | 9, 10, 11 | mp2an 692 | . . . . 5 ⊢ ((1...𝐽) ∩ 𝐶) ∈ Fin | 
| 13 | hashcl 14395 | . . . . 5 ⊢ (((1...𝐽) ∩ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0) | |
| 14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0 | 
| 15 | 14 | nn0zi 12642 | . . 3 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ | 
| 16 | difss 4136 | . . . . . 6 ⊢ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽) | |
| 17 | ssfi 9213 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∖ 𝐶) ∈ Fin) | |
| 18 | 9, 16, 17 | mp2an 692 | . . . . 5 ⊢ ((1...𝐽) ∖ 𝐶) ∈ Fin | 
| 19 | hashcl 14395 | . . . . 5 ⊢ (((1...𝐽) ∖ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0) | |
| 20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0 | 
| 21 | 20 | nn0zi 12642 | . . 3 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ | 
| 22 | zsubcl 12659 | . . 3 ⊢ (((♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ ∧ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ) | |
| 23 | 15, 21, 22 | mp2an 692 | . 2 ⊢ ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) ∈ ℤ | 
| 24 | 8, 23 | eqeltrdi 2849 | 1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 + caddc 11158 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ...cfz 13547 ♯chash 14369 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 | 
| This theorem is referenced by: ballotlemfc0 34495 ballotlemfcc 34496 ballotlemodife 34500 ballotlemic 34509 ballotlem1c 34510 ballotlemfrceq 34531 ballotlemfrcn0 34532 | 
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