ballotlemic - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemic		Structured version Visualization version GIF version

Theorem ballotlemic 34509

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 4131	. . . 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 34504	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
12	11	simpld 494	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
13		elfznn 13593	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℕ)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℕ)
15	14	adantr 480	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ ℕ)
16	1, 2, 3, 4, 5, 8, 9, 10	ballotlemi1 34505	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
17		eluz2b3 12964	. . . . . 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 12965	. . . . 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 480	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
22		elnnuz 12922	. . . . . . 7 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ ↔ ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
23	22	biimpi 216	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ → ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
24		eluzfz1 13571	. . . . . 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 480	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
27		1nn 12277	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℕ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfp1 34494	. . . . . . . . 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 494	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)))
31	30	imp 406	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1))
32		1m1e0 12338	. . . . . . . . . 10 ⊢ (1 − 1) = 0
33	32	fveq2i 6909	. . . . . . . . 9 ⊢ ((𝐹‘𝐶)‘(1 − 1)) = ((𝐹‘𝐶)‘0)
34	33	oveq1i 7441	. . . . . . . 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 34498	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0)
37	6, 36	syl 17	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘0) = 0)
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘0) = 0)
39	38	oveq1d 7446	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘0) − 1) = (0 − 1))
40	31, 35, 39	3eqtrrd 2782	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (0 − 1) = ((𝐹‘𝐶)‘1))
41		0le1 11786	. . . . . . 7 ⊢ 0 ≤ 1
42		0re 11263	. . . . . . . 8 ⊢ 0 ∈ ℝ
43		1re 11261	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		suble0 11777	. . . . . . . 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 231	. . . . . 6 ⊢ (0 − 1) ≤ 0
47	40, 46	eqbrtrrdi 5183	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) ≤ 0)
48	47	adantr 480	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘1) ≤ 0)
49		fveq2 6906	. . . . . 6 ⊢ (𝑖 = 1 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘1))
50	49	breq1d 5153	. . . . 5 ⊢ (𝑖 = 1 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘1) ≤ 0))
51	50	rspcev 3622	. . . 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 11785	. . . . 5 ⊢ 0 < 1
54		1p0e1 12390	. . . . . 6 ⊢ (1 + 0) = 1
55	1, 2, 3, 4, 5, 6, 14	ballotlemfp1 34494	. . . . . . . . . 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 494	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ (𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1)))
57	56	imp 406	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1))
58	11	simprd 495	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
59	58	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
60	57, 59	eqtr3d 2779	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1) = 0)
61	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
62	14	nnzd 12640	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
63	62	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (𝐼‘𝐶) ∈ ℤ)
64		1zzd 12648	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℤ)
65	63, 64	zsubcld 12727	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℤ)
66	1, 2, 3, 4, 5, 61, 65	ballotlemfelz 34493	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℤ)
67	66	zcnd 12723	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℂ)
68		1cnd 11256	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℂ)
69		0cnd 11254	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 0 ∈ ℂ)
70	67, 68, 69	subaddd 11638	. . . . . . 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 232	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → (1 + 0) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
72	54, 71	eqtr3id 2791	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ (𝐼‘𝐶) ∈ 𝐶) → 1 = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
73	53, 72	breqtrid 5180	. . . 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 34495	. 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 34508	. . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 ∖ cdif 3948 ∩ cin 3950 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 ♯chash 14369

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370

This theorem is referenced by: ballotlem7 34538

W3C validator