Theorem ballotlemodife 32464

Description: Elements of (𝑂 ∖ 𝐸). (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

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)}

Assertion

Ref	Expression
ballotlemodife	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂,𝑐   𝐹,𝑐,𝑖   𝐶,𝑖

Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑐)   𝐸(𝑥,𝑖,𝑐)   𝐹(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemodife

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3897	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸))
2		df-or 845	. . . 4 ⊢ (((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂) ∨ (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) ↔ (¬ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂) → (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
3		pm3.24 403	. . . . 5 ⊢ ¬ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂)
4	3	a1bi 363	. . . 4 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)) ↔ (¬ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂) → (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
5	2, 4	bitr4i 277	. . 3 ⊢ (((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂) ∨ (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) ↔ (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
6		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
7		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
8		ballotth.o	. . . . . . 7 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
9		ballotth.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10		ballotth.f	. . . . . . 7 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
11		ballotth.e	. . . . . . 7 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
12	6, 7, 8, 9, 10, 11	ballotleme 32463	. . . . . 6 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
13	12	notbii 320	. . . . 5 ⊢ (¬ 𝐶 ∈ 𝐸 ↔ ¬ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
14	13	anbi2i 623	. . . 4 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
15		ianor 979	. . . . 5 ⊢ (¬ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)) ↔ (¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
16	15	anbi2i 623	. . . 4 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) ↔ (𝐶 ∈ 𝑂 ∧ (¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
17		andi 1005	. . . 4 ⊢ ((𝐶 ∈ 𝑂 ∧ (¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) ↔ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂) ∨ (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
18	14, 16, 17	3bitri 297	. . 3 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸) ↔ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂) ∨ (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
19		fz1ssfz0 13352	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
21	20	sseld 3920	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑂 → (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ (0...(𝑀 + 𝑁))))
22	21	imdistani 569	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (0...(𝑀 + 𝑁))))
23		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶 ∈ 𝑂)
24		elfzelz 13256	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
25	24	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
26	6, 7, 8, 9, 10, 23, 25	ballotlemfelz 32457	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℤ)
27	26	zred 12426	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℝ)
28	27	sbimi 2077	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹‘𝐶)‘𝑗) ∈ ℝ)
29		sban 2083	. . . . . . . . . 10 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30		sbv 2091	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ↔ 𝐶 ∈ 𝑂)
31		clelsb1 2866	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑖 ∈ (0...(𝑀 + 𝑁)))
32	30, 31	anbi12i 627	. . . . . . . . . 10 ⊢ (([𝑖 / 𝑗]𝐶 ∈ 𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (0...(𝑀 + 𝑁))))
33	29, 32	bitri 274	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (0...(𝑀 + 𝑁))))
34		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑗((𝐹‘𝐶)‘𝑖) ∈ ℝ
35		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝐶)‘𝑗) = ((𝐹‘𝐶)‘𝑖))
36	35	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (((𝐹‘𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹‘𝐶)‘𝑖) ∈ ℝ))
37	34, 36	sbiev 2309	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗]((𝐹‘𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹‘𝐶)‘𝑖) ∈ ℝ)
38	28, 33, 37	3imtr3i 291	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑖) ∈ ℝ)
39	22, 38	syl 17	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑖) ∈ ℝ)
40		0red 10978	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
41	39, 40	lenltd 11121	. . . . . 6 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
42	41	rexbidva 3225	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
43		rexnal 3169	. . . . 5 ⊢ (∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹‘𝐶)‘𝑖) ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))
44	42, 43	bitrdi 287	. . . 4 ⊢ (𝐶 ∈ 𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
45	44	pm5.32i 575	. . 3 ⊢ ((𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0) ↔ (𝐶 ∈ 𝑂 ∧ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
46	5, 18, 45	3bitr4i 303	. 2 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))
47	1, 46	bitri 274	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539  [wsb 2067   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010   − cmin 11205   / cdiv 11632  ℕcn 11973  ℤcz 12319  ...cfz 13239  ♯chash 14044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045

This theorem is referenced by:  ballotlem5  32466  ballotlemrc  32497