Theorem ballotfilemodife 13158

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

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfi.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfi.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘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
ballotfilemodife	⊢ (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 ballotfilemodife

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3222	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸))
2		ballotth.m	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℕ
3		ballotth.n	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ
4		ballotfi.o	. . . . . . . . . 10 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
5		ballotfi.p	. . . . . . . . . 10 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
6		ballotth.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
7		ballotth.e	. . . . . . . . . 10 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
8	2, 3, 4, 5, 6, 7	ballotfileme 13157	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
9	8	baib 927	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 → (𝐶 ∈ 𝐸 ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
10		fveq2 5672	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑗))
11	10	breq2d 4123	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (0 < ((𝐹‘𝐶)‘𝑖) ↔ 0 < ((𝐹‘𝐶)‘𝑗)))
12	11	cbvralv 2780	. . . . . . . 8 ⊢ (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖) ↔ ∀𝑗 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑗))
13	9, 12	bitrdi 196	. . . . . . 7 ⊢ (𝐶 ∈ 𝑂 → (𝐶 ∈ 𝐸 ↔ ∀𝑗 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑗)))
14		0z 9590	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		fz1ssfz0 10455	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
16	15	sseli 3236	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ (0...(𝑀 + 𝑁)))
17		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶 ∈ 𝑂)
18		elfzelz 10362	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
19	18	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝑗 ∈ ℤ)
20	2, 3, 4, 5, 6, 17, 19	ballotfilemfelz 13151	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℤ)
21	16, 20	sylan2 286	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝑗) ∈ ℤ)
22		zltnle 9625	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑗) ∈ ℤ) → (0 < ((𝐹‘𝐶)‘𝑗) ↔ ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
23	14, 21, 22	sylancr 414	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → (0 < ((𝐹‘𝐶)‘𝑗) ↔ ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
24	23	ralbidva 2540	. . . . . . 7 ⊢ (𝐶 ∈ 𝑂 → (∀𝑗 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑗) ↔ ∀𝑗 ∈ (1...(𝑀 + 𝑁)) ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
25	13, 24	bitrd 188	. . . . . 6 ⊢ (𝐶 ∈ 𝑂 → (𝐶 ∈ 𝐸 ↔ ∀𝑗 ∈ (1...(𝑀 + 𝑁)) ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
26	25	notbid 673	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (¬ 𝐶 ∈ 𝐸 ↔ ¬ ∀𝑗 ∈ (1...(𝑀 + 𝑁)) ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
27		1z 9605	. . . . . . 7 ⊢ 1 ∈ ℤ
28	2	nnzi 9600	. . . . . . . 8 ⊢ 𝑀 ∈ ℤ
29	3	nnzi 9600	. . . . . . . 8 ⊢ 𝑁 ∈ ℤ
30		zaddcl 9619	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
31	28, 29, 30	mp2an 426	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℤ
32		fzfig 10796	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
33	27, 31, 32	mp2an 426	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
34		zdcle 9656	. . . . . . . 8 ⊢ ((((𝐹‘𝐶)‘𝑗) ∈ ℤ ∧ 0 ∈ ℤ) → DECID ((𝐹‘𝐶)‘𝑗) ≤ 0)
35	21, 14, 34	sylancl 413	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → DECID ((𝐹‘𝐶)‘𝑗) ≤ 0)
36	35	ralrimiva 2617	. . . . . 6 ⊢ (𝐶 ∈ 𝑂 → ∀𝑗 ∈ (1...(𝑀 + 𝑁))DECID ((𝐹‘𝐶)‘𝑗) ≤ 0)
37		dfrex2fin 7163	. . . . . 6 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ ∀𝑗 ∈ (1...(𝑀 + 𝑁))DECID ((𝐹‘𝐶)‘𝑗) ≤ 0) → (∃𝑗 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑗) ≤ 0 ↔ ¬ ∀𝑗 ∈ (1...(𝑀 + 𝑁)) ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
38	33, 36, 37	sylancr 414	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (∃𝑗 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑗) ≤ 0 ↔ ¬ ∀𝑗 ∈ (1...(𝑀 + 𝑁)) ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
39	26, 38	bitr4d 191	. . . 4 ⊢ (𝐶 ∈ 𝑂 → (¬ 𝐶 ∈ 𝐸 ↔ ∃𝑗 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑗) ≤ 0))
40	10	breq1d 4121	. . . . 5 ⊢ (𝑖 = 𝑗 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝑗) ≤ 0))
41	40	cbvrexv 2781	. . . 4 ⊢ (∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ∃𝑗 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑗) ≤ 0)
42	39, 41	bitr4di 198	. . 3 ⊢ (𝐶 ∈ 𝑂 → (¬ 𝐶 ∈ 𝐸 ↔ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))
43	42	pm5.32i 454	. 2 ⊢ ((𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))
44	1, 43	bitri 184	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105  DECID wdc 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526   ∖ cdif 3210   ∩ cin 3212  𝒫 cpw 3671   class class class wbr 4111   ↦ cmpt 4173  ‘cfv 5354  (class class class)co 6052  Fincfn 6977  0cc0 8129  1c1 8130   + caddc 8132   < clt 8310   ≤ cle 8311   − cmin 8446   / cdiv 8948  ℕcn 9239  ℤcz 9579  ...cfz 10345  ♯chash 11142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-ihash 11143

This theorem is referenced by: (None)