ballotfilemgun - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ballotfilemgun		GIF version

Theorem ballotfilemgun 13251

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 3480	. . . . . 6 ⊢ (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈) = (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∪ ((𝐿...𝐾) ∩ 𝑈))
2	1	fveq2i 5696	. . . . 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 10410	. . . . . . . 8 ⊢ (𝐿 ∈ (𝐽...𝐾) → 𝐽 ∈ ℤ)
9	7, 8	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
10	7	elfzelzd 10412	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ ℤ)
11		peano2zm 9665	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (𝐿 − 1) ∈ ℤ)
12	10, 11	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝐿 − 1) ∈ ℤ)
13	3, 4, 5, 6, 9, 12	ballotfilemcinfz 13209	. . . . . 6 ⊢ (𝜑 → ((𝐽...(𝐿 − 1)) ∩ 𝑈) ∈ Fin)
14		elfzel2 10409	. . . . . . . 8 ⊢ (𝐿 ∈ (𝐽...𝐾) → 𝐾 ∈ ℤ)
15	7, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16	3, 4, 5, 6, 10, 15	ballotfilemcinfz 13209	. . . . . 6 ⊢ (𝜑 → ((𝐿...𝐾) ∩ 𝑈) ∈ Fin)
17	10	zred 9751	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ ℝ)
18	17	ltm1d 9256	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 − 1) < 𝐿)
19		fzdisj 10440	. . . . . . . . 9 ⊢ ((𝐿 − 1) < 𝐿 → ((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) = ∅)
20	18, 19	syl 14	. . . . . . . 8 ⊢ (𝜑 → ((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) = ∅)
21	20	ineq1d 3431	. . . . . . 7 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∩ 𝑈) = (∅ ∩ 𝑈))
22		inindir 3449	. . . . . . 7 ⊢ (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∩ 𝑈) = (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∩ ((𝐿...𝐾) ∩ 𝑈))
23		0in 3558	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
24	21, 22, 23	3eqtr3g 2294	. . . . . 6 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∩ 𝑈) ∩ ((𝐿...𝐾) ∩ 𝑈)) = ∅)
25		hashun 11228	. . . . . 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 1278	. . . . 5 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∩ 𝑈) ∪ ((𝐿...𝐾) ∩ 𝑈))) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))))
27	2, 26	eqtrid 2283	. . . 4 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) + (♯‘((𝐿...𝐾) ∩ 𝑈))))
28		difundir 3484	. . . . . 6 ⊢ (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈) = (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈))
29	28	fveq2i 5696	. . . . 5 ⊢ (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈)))
30	3, 4, 5, 6, 9, 12	ballotfilemdifcfz 13210	. . . . . 6 ⊢ (𝜑 → ((𝐽...(𝐿 − 1)) ∖ 𝑈) ∈ Fin)
31	3, 4, 5, 6, 10, 15	ballotfilemdifcfz 13210	. . . . . 6 ⊢ (𝜑 → ((𝐿...𝐾) ∖ 𝑈) ∈ Fin)
32	20	difeq1d 3346	. . . . . . 7 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∖ 𝑈) = (∅ ∖ 𝑈))
33		difindir 3486	. . . . . . 7 ⊢ (((𝐽...(𝐿 − 1)) ∩ (𝐿...𝐾)) ∖ 𝑈) = (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∩ ((𝐿...𝐾) ∖ 𝑈))
34		0dif 3597	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
35	32, 33, 34	3eqtr3g 2294	. . . . . 6 ⊢ (𝜑 → (((𝐽...(𝐿 − 1)) ∖ 𝑈) ∩ ((𝐿...𝐾) ∖ 𝑈)) = ∅)
36		hashun 11228	. . . . . 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 1278	. . . . 5 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∖ 𝑈) ∪ ((𝐿...𝐾) ∖ 𝑈))) = ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈))))
38	29, 37	eqtrid 2283	. . . 4 ⊢ (𝜑 → (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈)) = ((♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) + (♯‘((𝐿...𝐾) ∖ 𝑈))))
39	27, 38	oveq12d 6097	. . 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 11203	. . . . . 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 9605	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) ∈ ℂ)
43		hashcl 11203	. . . . . 6 ⊢ (((𝐿...𝐾) ∩ 𝑈) ∈ Fin → (♯‘((𝐿...𝐾) ∩ 𝑈)) ∈ ℕ₀)
44	16, 43	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∩ 𝑈)) ∈ ℕ₀)
45	44	nn0cnd 9605	. . . 4 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∩ 𝑈)) ∈ ℂ)
46		hashcl 11203	. . . . . 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 9605	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈)) ∈ ℂ)
49		hashcl 11203	. . . . . 6 ⊢ (((𝐿...𝐾) ∖ 𝑈) ∈ Fin → (♯‘((𝐿...𝐾) ∖ 𝑈)) ∈ ℕ₀)
50	31, 49	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∖ 𝑈)) ∈ ℕ₀)
51	50	nn0cnd 9605	. . . 4 ⊢ (𝜑 → (♯‘((𝐿...𝐾) ∖ 𝑈)) ∈ ℂ)
52	42, 45, 48, 51	addsub4d 8678	. . 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 2271	. 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 2239	. . . 4 ⊢ (𝜑 → (𝐽...𝐾) = (𝐽...𝐾))
63	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 9, 15, 62	ballotfilemgval 13250	. . 3 ⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((♯‘((𝐽...𝐾) ∩ 𝑈)) − (♯‘((𝐽...𝐾) ∖ 𝑈))))
64		fzsplit3 10441	. . . . . . 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 3431	. . . . 5 ⊢ (𝜑 → ((𝐽...𝐾) ∩ 𝑈) = (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈))
67	66	fveq2d 5697	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...𝐾) ∩ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)))
68	65	difeq1d 3346	. . . . 5 ⊢ (𝜑 → ((𝐽...𝐾) ∖ 𝑈) = (((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))
69	68	fveq2d 5697	. . . 4 ⊢ (𝜑 → (♯‘((𝐽...𝐾) ∖ 𝑈)) = (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈)))
70	67, 69	oveq12d 6097	. . 3 ⊢ (𝜑 → ((♯‘((𝐽...𝐾) ∩ 𝑈)) − (♯‘((𝐽...𝐾) ∖ 𝑈))) = ((♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) − (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))))
71	63, 70	eqtrd 2271	. 2 ⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∩ 𝑈)) − (♯‘(((𝐽...(𝐿 − 1)) ∪ (𝐿...𝐾)) ∖ 𝑈))))
72		eqidd 2239	. . . 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 13250	. . 3 ⊢ (𝜑 → (𝑈 ↑ (𝐽...(𝐿 − 1))) = ((♯‘((𝐽...(𝐿 − 1)) ∩ 𝑈)) − (♯‘((𝐽...(𝐿 − 1)) ∖ 𝑈))))
74		eqidd 2239	. . . 4 ⊢ (𝜑 → (𝐿...𝐾) = (𝐿...𝐾))
75	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 10, 15, 74	ballotfilemgval 13250	. . 3 ⊢ (𝜑 → (𝑈 ↑ (𝐿...𝐾)) = ((♯‘((𝐿...𝐾) ∩ 𝑈)) − (♯‘((𝐿...𝐾) ∖ 𝑈))))
76	73, 75	oveq12d 6097	. 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 2281	1 ⊢ (𝜑 → (𝑈 ↑ (𝐽...𝐾)) = ((𝑈 ↑ (𝐽...(𝐿 − 1))) + (𝑈 ↑ (𝐿...𝐾))))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1402 ∈ wcel 2209 ∀wral 2528 {crab 2532 ∖ cdif 3217 ∪ cun 3218 ∩ cin 3219 ∅c0 3520 ifcif 3638 𝒫 cpw 3688 class class class wbr 4128 ↦ cmpt 4190 “ cima 4775 ‘cfv 5375 (class class class)co 6079 ∈ cmpo 6081 Fincfn 7016 infcinf 7317 ℝcr 8172 0cc0 8173 1c1 8174 + caddc 8176 < clt 8354 ≤ cle 8355 − cmin 8491 / cdiv 8996 ℕcn 9287 ℕ₀cn0 9546 ℤcz 9627 ...cfz 10394 ♯chash 11197

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 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 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-addass 8275 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289

This theorem 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-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 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-frec 6656 df-1o 6681 df-oadd 6685 df-er 6801 df-en 7017 df-dom 7018 df-fin 7019 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-inn 9288 df-n0 9547 df-z 9628 df-uz 9905 df-fz 10395 df-ihash 11198

This theorem is referenced by: ballotfilemfrceq 13255

W3C validator