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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemieq | Structured version Visualization version GIF version |
Description: If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-Apr-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, < )) |
ballotth.s | ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) |
Ref | Expression |
---|---|
ballotlemieq | ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑆‘𝐶) = (𝑆‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . . . . 6 ⊢ (((𝐼‘𝐶) = (𝐼‘𝐷) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝐶) = (𝐼‘𝐷)) | |
2 | 1 | breq2d 5042 | . . . . 5 ⊢ (((𝐼‘𝐶) = (𝐼‘𝐷) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝑖 ≤ (𝐼‘𝐶) ↔ 𝑖 ≤ (𝐼‘𝐷))) |
3 | 1 | oveq1d 7150 | . . . . . 6 ⊢ (((𝐼‘𝐶) = (𝐼‘𝐷) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝐶) + 1) = ((𝐼‘𝐷) + 1)) |
4 | 3 | oveq1d 7150 | . . . . 5 ⊢ (((𝐼‘𝐶) = (𝐼‘𝐷) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐷) + 1) − 𝑖)) |
5 | 2, 4 | ifbieq1d 4448 | . . . 4 ⊢ (((𝐼‘𝐶) = (𝐼‘𝐷) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) = if(𝑖 ≤ (𝐼‘𝐷), (((𝐼‘𝐷) + 1) − 𝑖), 𝑖)) |
6 | 5 | mpteq2dva 5125 | . . 3 ⊢ ((𝐼‘𝐶) = (𝐼‘𝐷) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐷), (((𝐼‘𝐷) + 1) − 𝑖), 𝑖))) |
7 | 6 | 3ad2ant3 1132 | . 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐷), (((𝐼‘𝐷) + 1) − 𝑖), 𝑖))) |
8 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
9 | ballotth.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
10 | ballotth.o | . . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
11 | ballotth.p | . . . 4 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) | |
12 | ballotth.f | . . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) | |
13 | ballotth.e | . . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} | |
14 | ballotth.mgtn | . . . 4 ⊢ 𝑁 < 𝑀 | |
15 | ballotth.i | . . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) | |
16 | ballotth.s | . . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) | |
17 | 8, 9, 10, 11, 12, 13, 14, 15, 16 | ballotlemsval 31876 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))) |
18 | 17 | 3ad2ant1 1130 | . 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))) |
19 | 8, 9, 10, 11, 12, 13, 14, 15, 16 | ballotlemsval 31876 | . . 3 ⊢ (𝐷 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐷) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐷), (((𝐼‘𝐷) + 1) − 𝑖), 𝑖))) |
20 | 19 | 3ad2ant2 1131 | . 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑆‘𝐷) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐷), (((𝐼‘𝐷) + 1) − 𝑖), 𝑖))) |
21 | 7, 18, 20 | 3eqtr4d 2843 | 1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑆‘𝐶) = (𝑆‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ifcif 4425 𝒫 cpw 4497 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 ℤcz 11969 ...cfz 12885 ♯chash 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 |
This theorem is referenced by: ballotlemrinv0 31900 |
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