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Theorem ballotfilemgun 13212

Description: A property of the defined ↑ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)

**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, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
ballotlemg	⊢ ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
ballotfilemgun.1	⊢ (𝜑 → 𝑈 ∈ 𝑂)
ballotfilemgun.2	⊢ (𝜑 → 𝐿 ∈ (𝐽...𝐾))

**Assertion**
Ref	Expression
ballotfilemgun	⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((𝑈 ↑ (𝐽...(𝐿 − 1))) + (𝑈 ↑ (𝐿...𝐾))))

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 𝑖,𝐸,𝑘 𝑘,𝐼,𝑐 𝐸,𝑐 𝑖,𝐼,𝑐 𝑘,𝐽 𝑆,𝑘,𝑖,𝑐 𝑅,𝑖 𝑣,𝑢,𝐼 𝑢,𝐽,𝑣 𝑢,𝑅,𝑣 𝑢,𝑆,𝑣 𝑢,𝑈,𝑣 𝑢,𝑂,𝑣 𝑢,𝐾,𝑣 𝑢,𝐿,𝑣 𝑘,𝐿 Allowed substitution hints: 𝜑(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝑃(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝑅(𝑥,𝑘,𝑐) 𝑆(𝑥) 𝑈(𝑥,𝑖,𝑘,𝑐) 𝐸(𝑥,𝑣,𝑢) ↑ (𝑥,𝑣,𝑢,𝑖,𝑘,𝑐) 𝐹(𝑥,𝑣,𝑢) 𝐼(𝑥) 𝐽(𝑥,𝑖,𝑐) 𝐾(𝑥,𝑖,𝑘,𝑐) 𝐿(𝑥,𝑖,𝑐) 𝑀(𝑥,𝑣,𝑢) 𝑁(𝑥,𝑣,𝑢) 𝑂(𝑥)

**Proof of Theorem ballotfilemgun**
Step	Hyp	Ref	Expression
1		indir 3474	. . . . . 6 ⊢ (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈) = (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∪ ((𝐿...𝐾) ∩ 𝑈))
2	1	fveq2i 5678	. . . . 5 ⊢ (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∩ 𝑈) ∪ ((𝐿...𝐾) ∩ 𝑈)))
3		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
4		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
5		ballotfilem.o	. . . . . . 7 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
6		ballotfilemgun.1	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑂)
7		ballotfilemgun.2	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐽...𝐾))
8		elfzel1 10377	. . . . . . . 8 ⊢ (𝐿 ∈ (𝐽...𝐾) → 𝐽 ∈ ℤ)
9	7, 8	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
10	7	elfzelzd 10379	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℤ)
11		peano2zm 9632	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (𝐿 − 1) ∈ ℤ)
12	10, 11	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝐿 − 1) ∈ ℤ)
13	3, 4, 5, 6, 9, 12	ballotfilemcinfz 13170	. . . . . 6 ⊢ (𝜑 → ((𝐽...(𝐿 − 1)) ∩ 𝑈) ∈ Fin)
14		elfzel2 10376	. . . . . . . 8 ⊢ (𝐿 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ)
15	7, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16	3, 4, 5, 6, 10, 15	ballotfilemcinfz 13170	. . . . . 6 ⊢ (𝜑 → ((𝐿...𝐾) ∩ 𝑈) ∈ Fin)
17	10	zred 9718	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℝ)
18	17	ltm1d 9223	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 − 1) < 𝐿)
19		fzdisj 10406	. . . . . . . . 9 ⊢ ((𝐿 − 1) < 𝐿 → ((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) = ∅)
20	18, 19	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) = ∅)
21	20	ineq1d 3425	. . . . . . 7 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∩ 𝑈) = (∅ ∩ 𝑈))
22		inindir 3443	. . . . . . 7 ⊢ (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∩ 𝑈) = (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∩ ((𝐿...𝐾) ∩ 𝑈))
23		0in 3548	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
24	21, 22, 23	3eqtr3g 2290	. . . . . 6 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∩ ((𝐿...𝐾) ∩ 𝑈)) = ∅)
25		hashun 11194	. . . . . 6 ⊢ ((((𝐽...(𝐿 − 1)) ∩ 𝑈) ∈ Fin ∧ ((𝐿...𝐾) ∩ 𝑈) ∈ Fin ∧ (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∩ ((𝐿...𝐾) ∩ 𝑈)) = ∅) → (♯‘(((𝐽...(𝐿 − 1)) ∩ 𝑈) ∪ ((𝐿...𝐾) ∩ 𝑈))) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))))
26	13, 16, 24, 25	syl3anc 1274	. . . . 5 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∩ 𝑈) ∪ ((𝐿...𝐾) ∩ 𝑈))) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))))
27	2, 26	eqtrid 2279	. . . 4 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))))
28		difundir 3478	. . . . . 6 ⊢ (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈) = (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈))
29	28	fveq2i 5678	. . . . 5 ⊢ (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈)))
30	3, 4, 5, 6, 9, 12	ballotfilemdifcfz 13171	. . . . . 6 ⊢ (𝜑 → ((𝐽...(𝐿 − 1)) ∖ 𝑈) ∈ Fin)
31	3, 4, 5, 6, 10, 15	ballotfilemdifcfz 13171	. . . . . 6 ⊢ (𝜑 → ((𝐿...𝐾) ∖ 𝑈) ∈ Fin)
32	20	difeq1d 3340	. . . . . . 7 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∖ 𝑈) = (∅ ∖ 𝑈))
33		difindir 3480	. . . . . . 7 ⊢ (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∖ 𝑈) = (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∩ ((𝐿...𝐾) ∖ 𝑈))
34		0dif 3584	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
35	32, 33, 34	3eqtr3g 2290	. . . . . 6 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∩ ((𝐿...𝐾) ∖ 𝑈)) = ∅)
36		hashun 11194	. . . . . 6 ⊢ ((((𝐽...(𝐿 − 1)) ∖ 𝑈) ∈ Fin ∧ ((𝐿...𝐾) ∖ 𝑈) ∈ Fin ∧ (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∩ ((𝐿...𝐾) ∖ 𝑈)) = ∅) → (♯‘(((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈))) = ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈))))
37	30, 31, 35, 36	syl3anc 1274	. . . . 5 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈))) = ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈))))
38	29, 37	eqtrid 2279	. . . 4 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈)) = ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈))))
39	27, 38	oveq12d 6076	. . 3 ⊢ (𝜑 → ((♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) − (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))) = (((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))) − ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈)))))
40		hashcl 11169	. . . . . 6 ⊢ (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∈ Fin → (♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) ∈ ℕ₀)
41	13, 40	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) ∈ ℕ₀)
42	41	nn0cnd 9572	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) ∈ ℂ)
43		hashcl 11169	. . . . . 6 ⊢ (((𝐿...𝐾) ∩ 𝑈) ∈ Fin → (♯‘((𝐿...𝐾) ∩ 𝑈)) ∈ ℕ₀)
44	16, 43	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∩ 𝑈)) ∈ ℕ₀)
45	44	nn0cnd 9572	. . . 4 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∩ 𝑈)) ∈ ℂ)
46		hashcl 11169	. . . . . 6 ⊢ (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∈ Fin → (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) ∈ ℕ₀)
47	30, 46	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) ∈ ℕ₀)
48	47	nn0cnd 9572	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) ∈ ℂ)
49		hashcl 11169	. . . . . 6 ⊢ (((𝐿...𝐾) ∖ 𝑈) ∈ Fin → (♯‘((𝐿...𝐾) ∖ 𝑈)) ∈ ℕ₀)
50	31, 49	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∖ 𝑈)) ∈ ℕ₀)
51	50	nn0cnd 9572	. . . 4 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∖ 𝑈)) ∈ ℂ)
52	42, 45, 48, 51	addsub4d 8647	. . 3 ⊢ (𝜑 → (((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))) − ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈)))) = (((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) − (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈))) + ((♯‘((𝐿...𝐾) ∩ 𝑈)) − (♯‘((𝐿...𝐾) ∖ 𝑈)))))
53	39, 52	eqtrd 2267	. 2 ⊢ (𝜑 → ((♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) − (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))) = (((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) − (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈))) + ((♯‘((𝐿...𝐾) ∩ 𝑈)) − (♯‘((𝐿...𝐾) ∖ 𝑈)))))
54		ballotfilem.p	. . . 4 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
55		ballotth.f	. . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
56		ballotth.e	. . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
57		ballotth.mgtn	. . . 4 ⊢ 𝑁 < 𝑀
58		ballotth.i	. . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
59		ballotth.s	. . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
60		ballotth.r	. . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
61		ballotlemg	. . . 4 ⊢ ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
62		eqidd 2235	. . . 4 ⊢ (𝜑 → (𝐽...𝐾) = (𝐽...𝐾))
63	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 9, 15, 62	ballotfilemgval 13211	. . 3 ⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((♯‘((𝐽...𝐾) ∩ 𝑈)) − (♯‘((𝐽...𝐾) ∖ 𝑈))))
64		fzsplit3 10407	. . . . . . 7 ⊢ (𝐿 ∈ (𝐽...𝐾) → (𝐽...𝐾) = ((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)))
65	7, 64	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐽...𝐾) = ((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)))
66	65	ineq1d 3425	. . . . 5 ⊢ (𝜑 → ((𝐽...𝐾) ∩ 𝑈) = (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈))
67	66	fveq2d 5679	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...𝐾) ∩ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)))
68	65	difeq1d 3340	. . . . 5 ⊢ (𝜑 → ((𝐽...𝐾) ∖ 𝑈) = (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))
69	68	fveq2d 5679	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...𝐾) ∖ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈)))
70	67, 69	oveq12d 6076	. . 3 ⊢ (𝜑 → ((♯‘((𝐽...𝐾) ∩ 𝑈)) − (♯‘((𝐽...𝐾) ∖ 𝑈))) = ((♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) − (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))))
71	63, 70	eqtrd 2267	. 2 ⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) − (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))))
72		eqidd 2235	. . . 4 ⊢ (𝜑 → (𝐽...(𝐿 − 1)) = (𝐽...(𝐿 − 1)))
73	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 9, 12, 72	ballotfilemgval 13211	. . 3 ⊢ (𝜑 → (𝑈 ↑ (𝐽...(𝐿 − 1))) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) − (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈))))
74		eqidd 2235	. . . 4 ⊢ (𝜑 → (𝐿...𝐾) = (𝐿...𝐾))
75	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 10, 15, 74	ballotfilemgval 13211	. . 3 ⊢ (𝜑 → (𝑈 ↑ (𝐿...𝐾)) = ((♯‘((𝐿...𝐾) ∩ 𝑈)) − (♯‘((𝐿...𝐾) ∖ 𝑈))))
76	73, 75	oveq12d 6076	. 2 ⊢ (𝜑 → ((𝑈 ↑ (𝐽...(𝐿 − 1))) + (𝑈 ↑ (𝐿...𝐾))) = (((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) − (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈))) + ((♯‘((𝐿...𝐾) ∩ 𝑈)) − (♯‘((𝐿...𝐾) ∖ 𝑈)))))
77	53, 71, 76	3eqtr4d 2277	1 ⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((𝑈 ↑ (𝐽...(𝐿 − 1))) + (𝑈 ↑ (𝐿...𝐾))))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2205 ∀wral 2522 {crab 2526 ∖ cdif 3211 ∪ cun 3212 ∩ cin 3213 ∅c0 3512 ifcif 3624 𝒫 cpw 3674 class class class wbr 4114 ↦ cmpt 4176 “ cima 4757 ‘cfv 5357 (class class class)co 6058 ∈ cmpo 6060 Fincfn 6988 infcinf 7287 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 < clt 8324 ≤ cle 8325 − cmin 8460 / cdiv 8963 ℕcn 9254 ℕ₀cn0 9513 ℤcz 9594 ...cfz 10361 ♯chash 11163

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-addass 8245 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-n0 9514 df-z 9595 df-uz 9872 df-fz 10362 df-ihash 11164

This theorem is referenced by: ballotfilemfrceq 13216

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