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Mathbox for Thierry Arnoux |
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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 34190 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶)) |
12 | imassrn 6070 | . . . 4 ⊢ ((𝑆‘𝐶) “ 𝐶) ⊆ ran (𝑆‘𝐶) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | ballotlemsf1o 34186 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶))) |
14 | 13 | simpld 493 | . . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁))) |
15 | f1ofo 6839 | . . . . 5 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁))) | |
16 | forn 6807 | . . . . 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 4026 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ 𝐶) ⊆ (1...(𝑀 + 𝑁))) |
19 | 11, 18 | eqsstrd 4012 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ⊆ (1...(𝑀 + 𝑁))) |
20 | f1of1 6831 | . . . . . . 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 4120 | . . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂) | |
23 | 1, 2, 3 | ballotlemelo 34160 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) |
24 | 22, 23 | sylib 217 | . . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀)) |
25 | 24 | simpld 493 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁))) |
26 | id 22 | . . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ (𝑂 ∖ 𝐸)) | |
27 | f1imaeng 9028 | . . . . . 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 1368 | . . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ 𝐶) ≈ 𝐶) |
29 | 11, 28 | eqbrtrd 5166 | . . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ≈ 𝐶) |
30 | hasheni 14334 | . . . 4 ⊢ ((𝑅‘𝐶) ≈ 𝐶 → (♯‘(𝑅‘𝐶)) = (♯‘𝐶)) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘(𝑅‘𝐶)) = (♯‘𝐶)) |
32 | 24 | simprd 494 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘𝐶) = 𝑀) |
33 | 31, 32 | eqtrd 2765 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (♯‘(𝑅‘𝐶)) = 𝑀) |
34 | 1, 2, 3 | ballotlemelo 34160 | . 2 ⊢ ((𝑅‘𝐶) ∈ 𝑂 ↔ ((𝑅‘𝐶) ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘(𝑅‘𝐶)) = 𝑀)) |
35 | 19, 33, 34 | sylanbrc 581 | 1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 {crab 3419 ∖ cdif 3938 ∩ cin 3940 ⊆ wss 3941 ifcif 4525 𝒫 cpw 4599 class class class wbr 5144 ↦ cmpt 5227 ◡ccnv 5672 ran crn 5674 “ cima 5676 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 ≈ cen 8954 infcinf 9459 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 ℤcz 12583 ...cfz 13511 ♯chash 14316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-hash 14317 |
This theorem is referenced by: ballotlemfrc 34199 ballotlemfrceq 34201 ballotlemfrcn0 34202 ballotlemrc 34203 |
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