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Mathbox for Thierry Arnoux |
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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 34471 | . 2 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶)))) |
9 | fzfi 14010 | . . . . . 6 ⊢ (1...𝐽) ∈ Fin | |
10 | inss1 4245 | . . . . . 6 ⊢ ((1...𝐽) ∩ 𝐶) ⊆ (1...𝐽) | |
11 | ssfi 9212 | . . . . . 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 14392 | . . . . 5 ⊢ (((1...𝐽) ∩ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℕ0 |
15 | 14 | nn0zi 12640 | . . 3 ⊢ (♯‘((1...𝐽) ∩ 𝐶)) ∈ ℤ |
16 | difss 4146 | . . . . . 6 ⊢ ((1...𝐽) ∖ 𝐶) ⊆ (1...𝐽) | |
17 | ssfi 9212 | . . . . . 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 14392 | . . . . 5 ⊢ (((1...𝐽) ∖ 𝐶) ∈ Fin → (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℕ0 |
21 | 20 | nn0zi 12640 | . . 3 ⊢ (♯‘((1...𝐽) ∖ 𝐶)) ∈ ℤ |
22 | zsubcl 12657 | . . 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 2847 | 1 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 1c1 11154 + caddc 11156 − cmin 11490 / cdiv 11918 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 ...cfz 13544 ♯chash 14366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 |
This theorem is referenced by: ballotlemfc0 34474 ballotlemfcc 34475 ballotlemodife 34479 ballotlemic 34488 ballotlem1c 34489 ballotlemfrceq 34510 ballotlemfrcn0 34511 |
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