Theorem ballotlemodife 34465

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 3915	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸))
2		df-or 848	. . . 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 34464	. . . . . 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 983	. . . . 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 1009	. . . 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 13544	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
21	20	sseld 3936	. . . . . . . . 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 13445	. . . . . . . . . . . . 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 34458	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℤ)
27	26	zred 12598	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℝ)
28	27	sbimi 2075	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹‘𝐶)‘𝑗) ∈ ℝ)
29		sban 2081	. . . . . . . . . 10 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30		sbv 2089	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ↔ 𝐶 ∈ 𝑂)
31		clelsb1 2855	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑖 ∈ (0...(𝑀 + 𝑁)))
32	30, 31	anbi12i 628	. . . . . . . . . 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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑗((𝐹‘𝐶)‘𝑖) ∈ ℝ
35		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝐶)‘𝑗) = ((𝐹‘𝐶)‘𝑖))
36	35	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (((𝐹‘𝐶)‘𝑗) ∈ ℝ ↔ ((𝐹‘𝐶)‘𝑖) ∈ ℝ))
37	34, 36	sbiev 2313	. . . . . . . . 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 11137	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
41	39, 40	lenltd 11280	. . . . . 6 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
42	41	rexbidva 3151	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
43		rexnal 3081	. . . . 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 847   = wceq 1540  [wsb 2065   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3396   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   < clt 11168   ≤ cle 11169   − cmin 11365   / cdiv 11795  ℕcn 12146  ℤcz 12489  ...cfz 13428  ♯chash 14255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-hash 14256

This theorem is referenced by:  ballotlem5  34467  ballotlemrc  34498