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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 29878 | . 2 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶)))) |
9 | fzfi 12633 | . . . . . 6 ⊢ (1...𝐽) ∈ Fin | |
10 | inss1 3795 | . . . . . 6 ⊢ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽) | |
11 | ssfi 8065 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∩ 𝐶) ∈ Fin) | |
12 | 9, 10, 11 | mp2an 704 | . . . . 5 ⊢ ((1...𝐽) ∩ 𝐶) ∈ Fin |
13 | hashcl 13009 | . . . . 5 ⊢ (((1...𝐽) ∩ 𝐶) ∈ Fin → (#‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (#‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0 |
15 | 14 | nn0zi 11279 | . . 3 ⊢ (#‘((1...𝐽) ∩ 𝐶)) ∈ ℤ |
16 | difss 3699 | . . . . . 6 ⊢ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽) | |
17 | ssfi 8065 | . . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽)) → ((1...𝐽) ∖ 𝐶) ∈ Fin) | |
18 | 9, 16, 17 | mp2an 704 | . . . . 5 ⊢ ((1...𝐽) ∖ 𝐶) ∈ Fin |
19 | hashcl 13009 | . . . . 5 ⊢ (((1...𝐽) ∖ 𝐶) ∈ Fin → (#‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (#‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0 |
21 | 20 | nn0zi 11279 | . . 3 ⊢ (#‘((1...𝐽) ∖ 𝐶)) ∈ ℤ |
22 | zsubcl 11296 | . . 3 ⊢ (((#‘((1...𝐽) ∩ 𝐶)) ∈ ℤ ∧ (#‘((1...𝐽) ∖ 𝐶)) ∈ ℤ) → ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ ℤ) | |
23 | 15, 21, 22 | mp2an 704 | . 2 ⊢ ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))) ∈ ℤ |
24 | 8, 23 | syl6eqel 2696 | 1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 + caddc 9818 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: ballotlemfc0 29881 ballotlemfcc 29882 ballotlemodife 29886 ballotlemic 29895 ballotlem1c 29896 ballotlemfrceq 29917 ballotlemfrcn0 29918 |
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