Theorem ballotfilemodife 13240

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 13236	. . . . . . . . 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 5695	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑗))
11	10	breq2d 4142	. . . . . . . . 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 9655	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		fz1ssfz0 10524	. . . . . . . . . . 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 10428	. . . . . . . . . . . 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 13230	. . . . . . . . . 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 9690	. . . . . . . . 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 9670	. . . . . . 7 ⊢ 1 ∈ ℤ
28	2	nnzi 9665	. . . . . . . 8 ⊢ 𝑀 ∈ ℤ
29	3	nnzi 9665	. . . . . . . 8 ⊢ 𝑁 ∈ ℤ
30		zaddcl 9684	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
31	28, 29, 30	mp2an 430	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℤ
32		fzfig 10867	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
33	27, 31, 32	mp2an 430	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
34		zdcle 9721	. . . . . . . 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 7208	. . . . . 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 4140	. . . . 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 4130   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  Fincfn 7022  0cc0 8179  1c1 8180   + caddc 8182   < clt 8360   ≤ cle 8361   − cmin 8497   / cdiv 9002  ℕcn 9304  ℤcz 9644  ...cfz 10411  ♯chash 11214

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-ihash 11215

This theorem is used by:  ballotfilem5  13242  ballotfilemrc  13274