Theorem ballotlemodife 34462

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 3986	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸))
2		df-or 847	. . . 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		pm3.24 402	. . . . 5 ⊢ ¬ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝑂)
4	3	a1bi 362	. . . 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 278	. . 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 34461	. . . . . 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 622	. . . 4 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))))
15		ianor 982	. . . . 5 ⊢ (¬ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)) ↔ (¬ 𝐶 ∈ 𝑂 ∨ ¬ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
16	15	anbi2i 622	. . . 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 1008	. . . 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 13680	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
21	20	sseld 4007	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑂 → (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ (0...(𝑀 + 𝑁))))
22	21	imdistani 568	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (0...(𝑀 + 𝑁))))
23		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶 ∈ 𝑂)
24		elfzelz 13584	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
26	6, 7, 8, 9, 10, 23, 25	ballotlemfelz 34455	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℤ)
27	26	zred 12747	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℝ)
28	27	sbimi 2074	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹‘𝐶)‘𝑗) ∈ ℝ)
29		sban 2080	. . . . . . . . . 10 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30		sbv 2088	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ↔ 𝐶 ∈ 𝑂)
31		clelsb1 2871	. . . . . . . . . . 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 275	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ (𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (0...(𝑀 + 𝑁))))
34		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑗((𝐹‘𝐶)‘𝑖) ∈ ℝ
35		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝐶)‘𝑗) = ((𝐹‘𝐶)‘𝑖))
36	35	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (((𝐹‘𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹‘𝐶)‘𝑖) ∈ ℝ))
37	34, 36	sbiev 2318	. . . . . . . . 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 11293	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
41	39, 40	lenltd 11436	. . . . . 6 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
42	41	rexbidva 3183	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
43		rexnal 3106	. . . . 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 574	. . 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 275	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537  [wsb 2064   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   − cmin 11520   / cdiv 11947  ℕcn 12293  ℤcz 12639  ...cfz 13567  ♯chash 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380

This theorem is referenced by:  ballotlem5  34464  ballotlemrc  34495