Theorem ballotfilemodife 13218

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 13214	. . . . . . . . 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 5690	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑗))
11	10	breq2d 4137	. . . . . . . . 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 9634	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		fz1ssfz0 10502	. . . . . . . . . . 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 10407	. . . . . . . . . . . 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 13208	. . . . . . . . . 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 9669	. . . . . . . . 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 9649	. . . . . . 7 ⊢ 1 ∈ ℤ
28	2	nnzi 9644	. . . . . . . 8 ⊢ 𝑀 ∈ ℤ
29	3	nnzi 9644	. . . . . . . 8 ⊢ 𝑁 ∈ ℤ
30		zaddcl 9663	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
31	28, 29, 30	mp2an 430	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℤ
32		fzfig 10845	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
33	27, 31, 32	mp2an 430	. . . . . 6 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
34		zdcle 9700	. . . . . . . 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 7198	. . . . . 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 4135	. . . . 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))

Syntax hints:  ¬ 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 3685   class class class wbr 4125   ↦ cmpt 4187  ‘cfv 5372  (class class class)co 6075  Fincfn 7012  0cc0 8169  1c1 8170   + caddc 8172   < clt 8350   ≤ cle 8351   − cmin 8487   / cdiv 8992  ℕcn 9283  ℤcz 9623  ...cfz 10390  ♯chash 11192

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-ihash 11193

This theorem is referenced by:  ballotfilem5  13220  ballotfilemrc  13252