ballotlemic - Mathbox for Thierry Arnoux

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Theorem ballotlemic 32043

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 4017	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
7	6	ad2antrr 726	. . 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 32038	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
12	11	simpld 498	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
13		elfznn 13027	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℕ)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℕ)
15	14	adantr 484	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ ℕ)
16	1, 2, 3, 4, 5, 8, 9, 10	ballotlemi1 32039	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
17		eluz2b3 12404	. . . . . 6 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) ↔ ((𝐼‘𝐶) ∈ ℕ ∧ (𝐼‘𝐶) ≠ 1))
18	15, 16, 17	sylanbrc 586	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ (ℤ_≥‘2))
19		uz2m1nn 12405	. . . . 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 484	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
22		elnnuz 12364	. . . . . . 7 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ ↔ ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
23	22	biimpi 219	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ → ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
24		eluzfz1 13005	. . . . . 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 484	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
27		1nn 11727	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℕ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfp1 32028	. . . . . . . . 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 498	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)))
31	30	imp 410	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1))
32		1m1e0 11788	. . . . . . . . . 10 ⊢ (1 − 1) = 0
33	32	fveq2i 6677	. . . . . . . . 9 ⊢ ((𝐹‘𝐶)‘(1 − 1)) = ((𝐹‘𝐶)‘0)
34	33	oveq1i 7180	. . . . . . . 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 32032	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0)
37	6, 36	syl 17	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘0) = 0)
38	37	adantr 484	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘0) = 0)
39	38	oveq1d 7185	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘0) − 1) = (0 − 1))
40	31, 35, 39	3eqtrrd 2778	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (0 − 1) = ((𝐹‘𝐶)‘1))
41		0le1 11241	. . . . . . 7 ⊢ 0 ≤ 1
42		0re 10721	. . . . . . . 8 ⊢ 0 ∈ ℝ
43		1re 10719	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		suble0 11232	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → ((0 − 1) ≤ 0 ↔ 0 ≤ 1))
45	42, 43, 44	mp2an 692	. . . . . . 7 ⊢ ((0 − 1) ≤ 0 ↔ 0 ≤ 1)
46	41, 45	mpbir 234	. . . . . 6 ⊢ (0 − 1) ≤ 0
47	40, 46	eqbrtrrdi 5070	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) ≤ 0)
48	47	adantr 484	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘1) ≤ 0)
49		fveq2 6674	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘1))
50	49	breq1d 5040	. . . . 5 ⊢ (𝑖 = 1 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘1) ≤ 0))
51	50	rspcev 3526	. . . 4 ⊢ ((1 ∈ (1...((𝐼‘𝐶) − 1)) ∧ ((𝐹‘𝐶)‘1) ≤ 0) → ∃𝑖 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑖) ≤ 0)
52	26, 48, 51	syl2anc 587	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑖 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑖) ≤ 0)
53		0lt1 11240	. . . . 5 ⊢ 0 < 1
54		1p0e1 11840	. . . . . 6 ⊢ (1 + 0) = 1
55	1, 2, 3, 4, 5, 6, 14	ballotlemfp1 32028	. . . . . . . . . 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 498	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ (𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1)))
57	56	imp 410	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1))
58	11	simprd 499	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
59	58	adantr 484	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
60	57, 59	eqtr3d 2775	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1) = 0)
61	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
62	14	nnzd 12167	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
63	62	adantr 484	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (𝐼‘𝐶) ∈ ℤ)
64		1zzd 12094	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℤ)
65	63, 64	zsubcld 12173	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℤ)
66	1, 2, 3, 4, 5, 61, 65	ballotlemfelz 32027	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℤ)
67	66	zcnd 12169	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℂ)
68		1cnd 10714	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℂ)
69		0cnd 10712	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 0 ∈ ℂ)
70	67, 68, 69	subaddd 11093	. . . . . . 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 235	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (1 + 0) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
72	54, 71	eqtr3id 2787	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
73	53, 72	breqtrid 5067	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 0 < ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
74	73	adantlr 715	. . 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 32029	. 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 32042	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
77	76	ad2antrr 726	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
78	75, 77	condan 818	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ∃wrex 3054 {crab 3057 ∖ cdif 3840 ∩ cin 3842 𝒫 cpw 4488 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7170 infcinf 8978 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 < clt 10753 ≤ cle 10754 − cmin 10948 / cdiv 11375 ℕcn 11716 2c2 11771 ℤcz 12062 ℤ_≥cuz 12324 ...cfz 12981 ♯chash 13782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-oadd 8135 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-hash 13783

This theorem is referenced by: ballotlem7 32072

W3C validator