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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemi1 | Structured version Visualization version GIF version |
Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.) |
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))))) |
ballotth.e | ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} |
ballotth.mgtn | ⊢ 𝑁 < 𝑀 |
ballotth.i | ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) |
Ref | Expression |
---|---|
ballotlemi1 | ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10443 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
2 | 1re 10441 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
3 | 1, 2 | resubcli 10751 | . . . . . 6 ⊢ (0 − 1) ∈ ℝ |
4 | 0lt1 10965 | . . . . . . 7 ⊢ 0 < 1 | |
5 | ltsub23 10923 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 − 1) < 0 ↔ (0 − 0) < 1)) | |
6 | 1, 2, 1, 5 | mp3an 1440 | . . . . . . . 8 ⊢ ((0 − 1) < 0 ↔ (0 − 0) < 1) |
7 | 0m0e0 11570 | . . . . . . . . 9 ⊢ (0 − 0) = 0 | |
8 | 7 | breq1i 4937 | . . . . . . . 8 ⊢ ((0 − 0) < 1 ↔ 0 < 1) |
9 | 6, 8 | bitr2i 268 | . . . . . . 7 ⊢ (0 < 1 ↔ (0 − 1) < 0) |
10 | 4, 9 | mpbi 222 | . . . . . 6 ⊢ (0 − 1) < 0 |
11 | 3, 10 | gtneii 10554 | . . . . 5 ⊢ 0 ≠ (0 − 1) |
12 | 11 | nesymi 3024 | . . . 4 ⊢ ¬ (0 − 1) = 0 |
13 | ballotth.m | . . . . . . . . 9 ⊢ 𝑀 ∈ ℕ | |
14 | ballotth.n | . . . . . . . . 9 ⊢ 𝑁 ∈ ℕ | |
15 | ballotth.o | . . . . . . . . 9 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
16 | ballotth.p | . . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) | |
17 | ballotth.f | . . . . . . . . 9 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) | |
18 | eldifi 3995 | . . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂) | |
19 | 1nn 11454 | . . . . . . . . . 10 ⊢ 1 ∈ ℕ | |
20 | 19 | a1i 11 | . . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℕ) |
21 | 13, 14, 15, 16, 17, 18, 20 | ballotlemfp1 31395 | . . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)) ∧ (1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1)))) |
22 | 21 | simpld 487 | . . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1))) |
23 | 22 | imp 398 | . . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)) |
24 | 1m1e0 11515 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
25 | 24 | fveq2i 6504 | . . . . . . . 8 ⊢ ((𝐹‘𝐶)‘(1 − 1)) = ((𝐹‘𝐶)‘0) |
26 | 25 | oveq1i 6988 | . . . . . . 7 ⊢ (((𝐹‘𝐶)‘(1 − 1)) − 1) = (((𝐹‘𝐶)‘0) − 1) |
27 | 26 | a1i 11 | . . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘(1 − 1)) − 1) = (((𝐹‘𝐶)‘0) − 1)) |
28 | 13, 14, 15, 16, 17 | ballotlemfval0 31399 | . . . . . . . . 9 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0) |
29 | 18, 28 | syl 17 | . . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘0) = 0) |
30 | 29 | adantr 473 | . . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘0) = 0) |
31 | 30 | oveq1d 6993 | . . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘0) − 1) = (0 − 1)) |
32 | 23, 27, 31 | 3eqtrrd 2819 | . . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (0 − 1) = ((𝐹‘𝐶)‘1)) |
33 | 32 | eqeq1d 2780 | . . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((0 − 1) = 0 ↔ ((𝐹‘𝐶)‘1) = 0)) |
34 | 12, 33 | mtbii 318 | . . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ¬ ((𝐹‘𝐶)‘1) = 0) |
35 | ballotth.e | . . . . . . 7 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} | |
36 | ballotth.mgtn | . . . . . . 7 ⊢ 𝑁 < 𝑀 | |
37 | ballotth.i | . . . . . . 7 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) | |
38 | 13, 14, 15, 16, 17, 35, 36, 37 | ballotlemiex 31405 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)) |
39 | 38 | simprd 488 | . . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0) |
40 | 39 | ad2antrr 713 | . . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) = 1) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0) |
41 | fveqeq2 6510 | . . . . 5 ⊢ ((𝐼‘𝐶) = 1 → (((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0 ↔ ((𝐹‘𝐶)‘1) = 0)) | |
42 | 41 | adantl 474 | . . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) = 1) → (((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0 ↔ ((𝐹‘𝐶)‘1) = 0)) |
43 | 40, 42 | mpbid 224 | . . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) = 1) → ((𝐹‘𝐶)‘1) = 0) |
44 | 34, 43 | mtand 803 | . 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ¬ (𝐼‘𝐶) = 1) |
45 | 44 | neqned 2974 | 1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 {crab 3092 ∖ cdif 3828 ∩ cin 3830 𝒫 cpw 4423 class class class wbr 4930 ↦ cmpt 5009 ‘cfv 6190 (class class class)co 6978 infcinf 8702 ℝcr 10336 0cc0 10337 1c1 10338 + caddc 10340 < clt 10476 − cmin 10672 / cdiv 11100 ℕcn 11441 ℤcz 11796 ...cfz 12711 ♯chash 13508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-sup 8703 df-inf 8704 df-dju 9126 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-hash 13509 |
This theorem is referenced by: ballotlemic 31410 |
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