Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemro | Structured version Visualization version GIF version |
Description: Range of 𝑅 is included in 𝑂. (Contributed by Thierry Arnoux, 17-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))) |
ballotth.r | ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) |
Ref | Expression |
---|---|
ballotlemro | ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
2 | ballotth.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | ballotth.o | . . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
4 | ballotth.p | . . . 4 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) | |
5 | ballotth.f | . . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) | |
6 | ballotth.e | . . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} | |
7 | ballotth.mgtn | . . . 4 ⊢ 𝑁 < 𝑀 | |
8 | ballotth.i | . . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) | |
9 | ballotth.s | . . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) | |
10 | ballotth.r | . . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ballotlemrval 31770 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶)) |
12 | imassrn 5934 | . . . 4 ⊢ ((𝑆‘𝐶) “ 𝐶) ⊆ ran (𝑆‘𝐶) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | ballotlemsf1o 31766 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶))) |
14 | 13 | simpld 497 | . . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁))) |
15 | f1ofo 6616 | . . . . 5 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁))) | |
16 | forn 6587 | . . . . 5 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) → ran (𝑆‘𝐶) = (1...(𝑀 + 𝑁))) | |
17 | 14, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ran (𝑆‘𝐶) = (1...(𝑀 + 𝑁))) |
18 | 12, 17 | sseqtrid 4018 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ 𝐶) ⊆ (1...(𝑀 + 𝑁))) |
19 | 11, 18 | eqsstrd 4004 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ⊆ (1...(𝑀 + 𝑁))) |
20 | f1of1 6608 | . . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁))) | |
21 | 14, 20 | syl 17 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁))) |
22 | eldifi 4102 | . . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂) | |
23 | 1, 2, 3 | ballotlemelo 31740 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) |
24 | 22, 23 | sylib 220 | . . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) |
25 | 24 | simpld 497 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁))) |
26 | id 22 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ (𝑂 ∖ 𝐸)) | |
27 | f1imaeng 8563 | . . . . . 6 ⊢ (((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ (𝑂 ∖ 𝐸)) → ((𝑆‘𝐶) “ 𝐶) ≈ 𝐶) | |
28 | 21, 25, 26, 27 | syl3anc 1367 | . . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ 𝐶) ≈ 𝐶) |
29 | 11, 28 | eqbrtrd 5080 | . . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ≈ 𝐶) |
30 | hasheni 13702 | . . . 4 ⊢ ((𝑅‘𝐶) ≈ 𝐶 → (♯‘(𝑅‘𝐶)) = (♯‘𝐶)) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘(𝑅‘𝐶)) = (♯‘𝐶)) |
32 | 24 | simprd 498 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘𝐶) = 𝑀) |
33 | 31, 32 | eqtrd 2856 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘(𝑅‘𝐶)) = 𝑀) |
34 | 1, 2, 3 | ballotlemelo 31740 | . 2 ⊢ ((𝑅‘𝐶) ∈ 𝑂 ↔ ((𝑅‘𝐶) ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘(𝑅‘𝐶)) = 𝑀)) |
35 | 19, 33, 34 | sylanbrc 585 | 1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ifcif 4466 𝒫 cpw 4538 class class class wbr 5058 ↦ cmpt 5138 ◡ccnv 5548 ran crn 5550 “ cima 5552 –1-1→wf1 6346 –onto→wfo 6347 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ≈ cen 8500 infcinf 8899 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − 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-rmo 3146 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-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-dju 9324 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-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-hash 13685 |
This theorem is referenced by: ballotlemfrc 31779 ballotlemfrceq 31781 ballotlemfrcn0 31782 ballotlemrc 31783 |
Copyright terms: Public domain | W3C validator |