Theorem ballotlemodife 34478

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 3972	. 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 34477	. . . . . 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 13659	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
21	20	sseld 3993	. . . . . . . . 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 13560	. . . . . . . . . . . . 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 34471	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℤ)
27	26	zred 12719	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℝ)
28	27	sbimi 2071	. . . . . . . . 9 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → [𝑖 / 𝑗]((𝐹‘𝐶)‘𝑗) ∈ ℝ)
29		sban 2077	. . . . . . . . . 10 ⊢ ([𝑖 / 𝑗](𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) ↔ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ∧ [𝑖 / 𝑗]𝑗 ∈ (0...(𝑀 + 𝑁))))
30		sbv 2085	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑗]𝐶 ∈ 𝑂 ↔ 𝐶 ∈ 𝑂)
31		clelsb1 2865	. . . . . . . . . . 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 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑗((𝐹‘𝐶)‘𝑖) ∈ ℝ
35		fveq2 6906	. . . . . . . . . . 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 2312	. . . . . . . . 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 11261	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → 0 ∈ ℝ)
41	39, 40	lenltd 11404	. . . . . 6 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
42	41	rexbidva 3174	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁)) ¬ 0 < ((𝐹‘𝐶)‘𝑖)))
43		rexnal 3097	. . . . 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 1536  [wsb 2061   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  {crab 3432   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604   class class class wbr 5147   ↦ cmpt 5230  ‘cfv 6562  (class class class)co 7430  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292   ≤ cle 11293   − cmin 11489   / cdiv 11917  ℕcn 12263  ℤcz 12610  ...cfz 13543  ♯chash 14365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366

This theorem is referenced by:  ballotlem5  34480  ballotlemrc  34511