ballotlemic - Mathbox for Thierry Arnoux

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Theorem ballotlemic 33493

Description: If the first vote is for B, the vote on the first tie is for A. (Contributed by Thierry Arnoux, 1-Dec-2016.)

**Hypotheses**
Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘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
ballotlemic	⊢ ((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 ballotlemic**
Step	Hyp	Ref	Expression
1		ballotth.m	. . 3 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . 3 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . 3 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . 3 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		eldifi 4125	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
7	6	ad2antrr 724	. . 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	ballotlemiex 33488	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
12	11	simpld 495	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
13		elfznn 13526	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℕ)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℕ)
15	14	adantr 481	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ ℕ)
16	1, 2, 3, 4, 5, 8, 9, 10	ballotlemi1 33489	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
17		eluz2b3 12902	. . . . . 6 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) ↔ ((𝐼‘𝐶) ∈ ℕ ∧ (𝐼‘𝐶) ≠ 1))
18	15, 16, 17	sylanbrc 583	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ (ℤ_≥‘2))
19		uz2m1nn 12903	. . . . 5 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) → ((𝐼‘𝐶) − 1) ∈ ℕ)
20	18, 19	syl 17	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
21	20	adantr 481	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
22		elnnuz 12862	. . . . . . 7 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ ↔ ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
23	22	biimpi 215	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ → ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
24		eluzfz1 13504	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
25	20, 23, 24	3syl 18	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
26	25	adantr 481	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
27		1nn 12219	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℕ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfp1 33478	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)) ∧ (1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1))))
30	29	simpld 495	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)))
31	30	imp 407	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1))
32		1m1e0 12280	. . . . . . . . . 10 ⊢ (1 − 1) = 0
33	32	fveq2i 6891	. . . . . . . . 9 ⊢ ((𝐹‘𝐶)‘(1 − 1)) = ((𝐹‘𝐶)‘0)
34	33	oveq1i 7415	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘(1 − 1)) − 1) = (((𝐹‘𝐶)‘0) − 1)
35	34	a1i 11	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘(1 − 1)) − 1) = (((𝐹‘𝐶)‘0) − 1))
36	1, 2, 3, 4, 5	ballotlemfval0 33482	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0)
37	6, 36	syl 17	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘0) = 0)
38	37	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘0) = 0)
39	38	oveq1d 7420	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘0) − 1) = (0 − 1))
40	31, 35, 39	3eqtrrd 2777	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (0 − 1) = ((𝐹‘𝐶)‘1))
41		0le1 11733	. . . . . . 7 ⊢ 0 ≤ 1
42		0re 11212	. . . . . . . 8 ⊢ 0 ∈ ℝ
43		1re 11210	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		suble0 11724	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ((0 − 1) ≤ 0 ↔ 0 ≤ 1))
45	42, 43, 44	mp2an 690	. . . . . . 7 ⊢ ((0 − 1) ≤ 0 ↔ 0 ≤ 1)
46	41, 45	mpbir 230	. . . . . 6 ⊢ (0 − 1) ≤ 0
47	40, 46	eqbrtrrdi 5187	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) ≤ 0)
48	47	adantr 481	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘1) ≤ 0)
49		fveq2 6888	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘1))
50	49	breq1d 5157	. . . . 5 ⊢ (𝑖 = 1 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘1) ≤ 0))
51	50	rspcev 3612	. . . 4 ⊢ ((1 ∈ (1...((𝐼‘𝐶) − 1)) ∧ ((𝐹‘𝐶)‘1) ≤ 0) → ∃𝑖 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑖) ≤ 0)
52	26, 48, 51	syl2anc 584	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑖 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑖) ≤ 0)
53		0lt1 11732	. . . . 5 ⊢ 0 < 1
54		1p0e1 12332	. . . . . 6 ⊢ (1 + 0) = 1
55	1, 2, 3, 4, 5, 6, 14	ballotlemfp1 33478	. . . . . . . . . 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))))
56	55	simpld 495	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ (𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1)))
57	56	imp 407	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1))
58	11	simprd 496	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
59	58	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
60	57, 59	eqtr3d 2774	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1) = 0)
61	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
62	14	nnzd 12581	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
63	62	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (𝐼‘𝐶) ∈ ℤ)
64		1zzd 12589	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℤ)
65	63, 64	zsubcld 12667	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℤ)
66	1, 2, 3, 4, 5, 61, 65	ballotlemfelz 33477	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℤ)
67	66	zcnd 12663	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℂ)
68		1cnd 11205	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℂ)
69		0cnd 11203	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 0 ∈ ℂ)
70	67, 68, 69	subaddd 11585	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1) = 0 ↔ (1 + 0) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1))))
71	60, 70	mpbid 231	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (1 + 0) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
72	54, 71	eqtr3id 2786	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
73	53, 72	breqtrid 5184	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 0 < ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
74	73	adantlr 713	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 0 < ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
75	1, 2, 3, 4, 5, 7, 21, 52, 74	ballotlemfc0 33479	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
76	1, 2, 3, 4, 5, 8, 9, 10	ballotlemimin 33492	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
77	76	ad2antrr 724	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
78	75, 77	condan 816	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3944 ∩ cin 3946 𝒫 cpw 4601 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 infcinf 9432 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ♯chash 14286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287

This theorem is referenced by: ballotlem7 33522

W3C validator