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 11987 | . . 3 ⊢ (𝐶 ∈ 𝑂 → 0 ∈ ℤ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ballotlemfval 31742 | . 2 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = ((♯‘((1...0) ∩ 𝐶)) − (♯‘((1...0) ∖ 𝐶)))) |
9 | fz10 12922 | . . . . . . . 8 ⊢ (1...0) = ∅ | |
10 | 9 | ineq1i 4184 | . . . . . . 7 ⊢ ((1...0) ∩ 𝐶) = (∅ ∩ 𝐶) |
11 | incom 4177 | . . . . . . 7 ⊢ (𝐶 ∩ ∅) = (∅ ∩ 𝐶) | |
12 | in0 4344 | . . . . . . 7 ⊢ (𝐶 ∩ ∅) = ∅ | |
13 | 10, 11, 12 | 3eqtr2i 2850 | . . . . . 6 ⊢ ((1...0) ∩ 𝐶) = ∅ |
14 | 13 | fveq2i 6667 | . . . . 5 ⊢ (♯‘((1...0) ∩ 𝐶)) = (♯‘∅) |
15 | hash0 13722 | . . . . 5 ⊢ (♯‘∅) = 0 | |
16 | 14, 15 | eqtri 2844 | . . . 4 ⊢ (♯‘((1...0) ∩ 𝐶)) = 0 |
17 | 9 | difeq1i 4094 | . . . . . . 7 ⊢ ((1...0) ∖ 𝐶) = (∅ ∖ 𝐶) |
18 | 0dif 4354 | . . . . . . 7 ⊢ (∅ ∖ 𝐶) = ∅ | |
19 | 17, 18 | eqtri 2844 | . . . . . 6 ⊢ ((1...0) ∖ 𝐶) = ∅ |
20 | 19 | fveq2i 6667 | . . . . 5 ⊢ (♯‘((1...0) ∖ 𝐶)) = (♯‘∅) |
21 | 20, 15 | eqtri 2844 | . . . 4 ⊢ (♯‘((1...0) ∖ 𝐶)) = 0 |
22 | 16, 21 | oveq12i 7162 | . . 3 ⊢ ((♯‘((1...0) ∩ 𝐶)) − (♯‘((1...0) ∖ 𝐶))) = (0 − 0) |
23 | 0m0e0 11751 | . . 3 ⊢ (0 − 0) = 0 | |
24 | 22, 23 | eqtri 2844 | . 2 ⊢ ((♯‘((1...0) ∩ 𝐶)) − (♯‘((1...0) ∖ 𝐶))) = 0 |
25 | 8, 24 | syl6eq 2872 | 1 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 ∖ cdif 3932 ∩ cin 3934 ∅c0 4290 𝒫 cpw 4538 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 − cmin 10864 / cdiv 11291 ℕcn 11632 ℤcz 11975 ...cfz 12886 ♯chash 13684 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 |
This theorem is referenced by: ballotlem4 31751 ballotlemi1 31755 ballotlemii 31756 ballotlemic 31759 ballotlem1c 31760 |
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