Theorem ballotfilemodife 13298

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

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.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 3229	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ¬ 𝐶 ∈ 𝐸))
2		ballotth.m	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℕ
3		ballotth.n	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ
4		ballotfilem.o	. . . . . . . . . 10 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
5		ballotfilem.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 13294	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
9	8	baib 931	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 → (𝐶 ∈ 𝐸 ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
10		fveq2 5698	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑗))
11	10	breq2d 4143	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (0 < ((𝐹‘𝐶)‘𝑖) ↔ 0 < ((𝐹‘𝐶)‘𝑗)))
12	11	cbvralv 2786	. . . . . . . 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 9666	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		fz1ssfz0 10541	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
16	15	sseli 3244	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ (0...(𝑀 + 𝑁)))
17		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → 𝐶 ∈ 𝑂)
18		elfzelz 10445	. . . . . . . . . . . 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 13288	. . . . . . . . . 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 9701	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑗) ∈ ℤ) → (0 < ((𝐹‘𝐶)‘𝑗) ↔ ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
23	14, 21, 22	sylancr 418	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → (0 < ((𝐹‘𝐶)‘𝑗) ↔ ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
24	23	ralbidva 2546	. . . . . . 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 677	. . . . 5 ⊢ (𝐶 ∈ 𝑂 → (¬ 𝐶 ∈ 𝐸 ↔ ¬ ∀𝑗 ∈ (1...(𝑀 + 𝑁)) ¬ ((𝐹‘𝐶)‘𝑗) ≤ 0))
27		1z 9681	. . . . . . 7 ⊢ 1 ∈ ℤ
28	2	nnzi 9676	. . . . . . . 8 ⊢ 𝑀 ∈ ℤ
29	3	nnzi 9676	. . . . . . . 8 ⊢ 𝑁 ∈ ℤ
30		zaddcl 9695	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
31	28, 29, 30	mp2an 430	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℤ
32		fzfig 10888	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
33	27, 31, 32	mp2an 430	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
34		zdcle 9732	. . . . . . . 8 ⊢ ((((𝐹‘𝐶)‘𝑗) ∈ ℤ ∧ 0 ∈ ℤ) → DECID ((𝐹‘𝐶)‘𝑗) ≤ 0)
35	21, 14, 34	sylancl 417	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑂 ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → DECID ((𝐹‘𝐶)‘𝑗) ≤ 0)
36	35	ralrimiva 2623	. . . . . 6 ⊢ (𝐶 ∈ 𝑂 → ∀𝑗 ∈ (1...(𝑀 + 𝑁))DECID ((𝐹‘𝐶)‘𝑗) ≤ 0)
37		dfrex2fin 7212	. . . . . 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 418	. . . . 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 4141	. . . . 5 ⊢ (𝑖 = 𝑗 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝑗) ≤ 0))
41	40	cbvrexv 2787	. . . 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 458	. 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))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532   ∖ cdif 3217   ∩ cin 3219  𝒫 cpw 3688   class class class wbr 4131   ↦ cmpt 4193  ‘cfv 5380  (class class class)co 6088  Fincfn 7026  0cc0 8185  1c1 8186   + caddc 8188   < clt 8366   ≤ cle 8367   − cmin 8504   / cdiv 9010  ℕcn 9313  ℤcz 9655  ...cfz 10428  ♯chash 11236

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-ihash 11237

This theorem is used by:  ballotfilem5  13300  ballotfilemrc  13332