ballotfilem1c - Intuitionistic Logic Explorer

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Theorem ballotfilem1c 13310

Description: If the first vote is for A, the vote on the first tie is for B. (Contributed by Thierry Arnoux, 4-Apr-2017.)

**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)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))

**Assertion**
Ref	Expression
ballotfilem1c	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ¬ (𝐼‘𝐶) ∈ 𝐶)

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐 𝑖,𝑀 𝑖,𝑁 𝑖,𝑂 𝑘,𝑀 𝑘,𝑁 𝑘,𝑂 𝑖,𝑐,𝐹,𝑘 𝐶,𝑖,𝑘 𝑖,𝐸,𝑘 𝐶,𝑘 𝑘,𝐼 𝑘,𝑐,𝐸 𝑖,𝐼 Allowed substitution hints: 𝐶(𝑥, 𝑐) 𝑃(𝑥, 𝑖, 𝑘, 𝑐) 𝐸(𝑥) 𝐹(𝑥) 𝐼(𝑥, 𝑐) 𝑀(𝑥) 𝑁(𝑥) 𝑂(𝑥)

**Proof of Theorem ballotfilem1c**
Step	Hyp	Ref	Expression
1		ballotth.m	. . 3 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . 3 ⊢ 𝑁 ∈ ℕ
3		ballotfilem.o	. . 3 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4		ballotfilem.p	. . 3 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		eldifi 3351	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
7	6	ad2antrr 492	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
8		ballotth.e	. . . . . . . . . 10 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
9		ballotth.mgtn	. . . . . . . . . 10 ⊢ 𝑁 < 𝑀
10		ballotth.i	. . . . . . . . . 10 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
11	1, 2, 3, 4, 5, 8, 9, 10	ballotfilemiex 13303	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
12	11	simpld 112	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
13		elfznn 10478	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℕ)
14	12, 13	syl 14	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℕ)
15	14	adantr 276	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ ℕ)
16	1, 2, 3, 4, 5, 8, 9, 10	ballotfilemii 13305	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
17		eluz2b3 10021	. . . . . 6 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) ↔ ((𝐼‘𝐶) ∈ ℕ ∧ (𝐼‘𝐶) ≠ 1))
18	15, 16, 17	sylanbrc 421	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ (ℤ_≥‘2))
19		uz2m1nn 10022	. . . . 5 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) → ((𝐼‘𝐶) − 1) ∈ ℕ)
20	18, 19	syl 14	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
21	20	adantr 276	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
22		fveq2 5698	. . . . 5 ⊢ (𝑖 = 1 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘1))
23	22	breq2d 4143	. . . 4 ⊢ (𝑖 = 1 → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘1)))
24		elnnuz 9976	. . . . . . 7 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ ↔ ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
25	24	biimpi 120	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ → ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
26		eluzfz1 10453	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
27	20, 25, 26	3syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
28	27	adantr 276	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
29		0le1 8817	. . . . . . 7 ⊢ 0 ≤ 1
30		1e0p1 9835	. . . . . . 7 ⊢ 1 = (0 + 1)
31	29, 30	breqtri 4156	. . . . . 6 ⊢ 0 ≤ (0 + 1)
32		1nn 9325	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
33	32	a1i 9	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℕ)
34	1, 2, 3, 4, 5, 6, 33	ballotfilemfp1 13290	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)) ∧ (1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1))))
35	34	simprd 114	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1)))
36	35	imp 124	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1))
37		1m1e0 9383	. . . . . . . . . 10 ⊢ (1 − 1) = 0
38	37	fveq2i 5701	. . . . . . . . 9 ⊢ ((𝐹‘𝐶)‘(1 − 1)) = ((𝐹‘𝐶)‘0)
39	38	oveq1i 6098	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘(1 − 1)) + 1) = (((𝐹‘𝐶)‘0) + 1)
40	39	a1i 9	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘(1 − 1)) + 1) = (((𝐹‘𝐶)‘0) + 1))
41	1, 2, 3, 4, 5	ballotfilemfval0 13294	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0)
42	6, 41	syl 14	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘0) = 0)
43	42	adantr 276	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘0) = 0)
44	43	oveq1d 6103	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘0) + 1) = (0 + 1))
45	36, 40, 44	3eqtrrd 2276	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (0 + 1) = ((𝐹‘𝐶)‘1))
46	31, 45	breqtrid 4168	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘1))
47	46	adantr 276	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘1))
48	23, 28, 47	rspcedvdw 2936	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑖 ∈ (1...((𝐼‘𝐶) − 1))0 ≤ ((𝐹‘𝐶)‘𝑖))
49		df-neg 8508	. . . . . 6 ⊢ -1 = (0 − 1)
50	1, 2, 3, 4, 5, 6, 14	ballotfilemfp1 13290	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((¬ (𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1)) ∧ ((𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1))))
51	50	simprd 114	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1)))
52	51	imp 124	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1))
53	11	simprd 114	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
54	53	adantr 276	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
55	52, 54	eqtr3d 2273	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1) = 0)
56		0cnd 8326	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 0 ∈ ℂ)
57		1cnd 8349	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℂ)
58	6	adantr 276	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
59	14	nnzd 9779	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
60	59	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → (𝐼‘𝐶) ∈ ℤ)
61		1zzd 9683	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℤ)
62	60, 61	zsubcld 9785	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℤ)
63	1, 2, 3, 4, 5, 58, 62	ballotfilemfelz 13289	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℤ)
64	63	zcnd 9781	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℂ)
65	56, 57, 64	subadd2d 8664	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((0 − 1) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ↔ (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1) = 0))
66	55, 65	mpbird 167	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → (0 − 1) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
67	49, 66	eqtrid 2283	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → -1 = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
68		neg1lt0 9422	. . . . 5 ⊢ -1 < 0
69	67, 68	eqbrtrrdi 4171	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) < 0)
70	69	adantlr 481	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) < 0)
71	1, 2, 3, 4, 5, 7, 21, 48, 70	ballotfilemfcc 13292	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
72	1, 2, 3, 4, 5, 8, 9, 10	ballotfilemimin 13308	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
73	72	ad2antrr 492	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
74	71, 73	pm2.65da 671	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ¬ (𝐼‘𝐶) ∈ 𝐶)

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∀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 7027 infcinf 7330 ℝcr 8185 0cc0 8186 1c1 8187 + caddc 8189 < clt 8367 ≤ cle 8368 − cmin 8505 -cneg 8506 / cdiv 9011 ℕcn 9314 2c2 9365 ℤcz 9656 ℤ_≥cuz 9938 ...cfz 10429 ♯chash 11237

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 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304

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-rmo 2536 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-po 4443 df-iso 4444 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-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-en 7028 df-dom 7029 df-fin 7030 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-fz 10430 df-fzo 10568 df-ihash 11238

This theorem is used by: ballotfilem7 13338

W3C validator