Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemfval0 | Structured version Visualization version GIF version |
Description: (𝐹‘𝐶) always starts counting at 0 . (Contributed by Thierry Arnoux, 25-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))))) |
Ref | Expression |
---|---|
ballotlemfval0 | ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0) |
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 | id 22 | . . 3 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ∈ 𝑂) | |
7 | 0zd 12067 | . . 3 ⊢ (𝐶 ∈ 𝑂 → 0 ∈ ℤ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ballotlemfval 32018 | . 2 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = ((♯‘((1...0) ∩ 𝐶)) − (♯‘((1...0) ∖ 𝐶)))) |
9 | fz10 13012 | . . . . . . . 8 ⊢ (1...0) = ∅ | |
10 | 9 | ineq1i 4097 | . . . . . . 7 ⊢ ((1...0) ∩ 𝐶) = (∅ ∩ 𝐶) |
11 | incom 4089 | . . . . . . 7 ⊢ (𝐶 ∩ ∅) = (∅ ∩ 𝐶) | |
12 | in0 4277 | . . . . . . 7 ⊢ (𝐶 ∩ ∅) = ∅ | |
13 | 10, 11, 12 | 3eqtr2i 2767 | . . . . . 6 ⊢ ((1...0) ∩ 𝐶) = ∅ |
14 | 13 | fveq2i 6671 | . . . . 5 ⊢ (♯‘((1...0) ∩ 𝐶)) = (♯‘∅) |
15 | hash0 13813 | . . . . 5 ⊢ (♯‘∅) = 0 | |
16 | 14, 15 | eqtri 2761 | . . . 4 ⊢ (♯‘((1...0) ∩ 𝐶)) = 0 |
17 | 9 | difeq1i 4007 | . . . . . . 7 ⊢ ((1...0) ∖ 𝐶) = (∅ ∖ 𝐶) |
18 | 0dif 4287 | . . . . . . 7 ⊢ (∅ ∖ 𝐶) = ∅ | |
19 | 17, 18 | eqtri 2761 | . . . . . 6 ⊢ ((1...0) ∖ 𝐶) = ∅ |
20 | 19 | fveq2i 6671 | . . . . 5 ⊢ (♯‘((1...0) ∖ 𝐶)) = (♯‘∅) |
21 | 20, 15 | eqtri 2761 | . . . 4 ⊢ (♯‘((1...0) ∖ 𝐶)) = 0 |
22 | 16, 21 | oveq12i 7176 | . . 3 ⊢ ((♯‘((1...0) ∩ 𝐶)) − (♯‘((1...0) ∖ 𝐶))) = (0 − 0) |
23 | 0m0e0 11829 | . . 3 ⊢ (0 − 0) = 0 | |
24 | 22, 23 | eqtri 2761 | . 2 ⊢ ((♯‘((1...0) ∩ 𝐶)) − (♯‘((1...0) ∖ 𝐶))) = 0 |
25 | 8, 24 | eqtrdi 2789 | 1 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 {crab 3057 ∖ cdif 3838 ∩ cin 3840 ∅c0 4209 𝒫 cpw 4485 ↦ cmpt 5107 ‘cfv 6333 (class class class)co 7164 0cc0 10608 1c1 10609 + caddc 10611 − cmin 10941 / cdiv 11368 ℕcn 11709 ℤcz 12055 ...cfz 12974 ♯chash 13775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-hash 13776 |
This theorem is referenced by: ballotlem4 32027 ballotlemi1 32031 ballotlemii 32032 ballotlemic 32035 ballotlem1c 32036 |
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