ballotlemgun - Mathbox for Thierry Arnoux

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

Theorem ballotlemgun 34562

Description: A property of the defined ↑ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.)

**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, < ))
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 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
ballotlemgun.1	⊢ (𝜑 → 𝑈 ∈ Fin)
ballotlemgun.2	⊢ (𝜑 → 𝑉 ∈ Fin)
ballotlemgun.3	⊢ (𝜑 → 𝑊 ∈ Fin)
ballotlemgun.4	⊢ (𝜑 → (𝑉 ∩ 𝑊) = ∅)

**Assertion**
Ref	Expression
ballotlemgun	⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)))

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 𝑆,𝑘,𝑖,𝑐 𝑅,𝑖 𝑣,𝑢,𝐼 𝑢,𝑅,𝑣 𝑢,𝑆,𝑣 𝑢,𝑈,𝑣 𝑢,𝑉,𝑣 𝑢,𝑊,𝑣 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) 𝑀(𝑥,𝑣,𝑢) 𝑁(𝑥,𝑣,𝑢) 𝑂(𝑥,𝑣,𝑢) 𝑉(𝑥,𝑖,𝑘,𝑐) 𝑊(𝑥,𝑖,𝑘,𝑐)

**Proof of Theorem ballotlemgun**
Step	Hyp	Ref	Expression
1		indir 4266	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))
2	1	fveq2i 6884	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈)))
3		ballotlemgun.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Fin)
4		infi 9279	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∩ 𝑈) ∈ Fin)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑉 ∩ 𝑈) ∈ Fin)
6		ballotlemgun.3	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
7		infi 9279	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∩ 𝑈) ∈ Fin)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∩ 𝑈) ∈ Fin)
9		ballotlemgun.4	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ∩ 𝑊) = ∅)
10	9	ineq1d 4199	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∩ 𝑈) = (∅ ∩ 𝑈))
11		inindir 4216	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈))
12		0in 4377	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
13	10, 11, 12	3eqtr3g 2794	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈)) = ∅)
14		hashun 14405	. . . . . 6 ⊢ (((𝑉 ∩ 𝑈) ∈ Fin ∧ (𝑊 ∩ 𝑈) ∈ Fin ∧ ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈)) = ∅) → (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
15	5, 8, 13, 14	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
16	2, 15	eqtrid 2783	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
17		difundir 4271	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))
18	17	fveq2i 6884	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈)))
19		diffi 9194	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ 𝑈) ∈ Fin)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
21		diffi 9194	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∖ 𝑈) ∈ Fin)
22	6, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∖ 𝑈) ∈ Fin)
23	9	difeq1d 4105	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∖ 𝑈) = (∅ ∖ 𝑈))
24		difindir 4273	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈))
25		0dif 4385	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
26	23, 24, 25	3eqtr3g 2794	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈)) = ∅)
27		hashun 14405	. . . . . 6 ⊢ (((𝑉 ∖ 𝑈) ∈ Fin ∧ (𝑊 ∖ 𝑈) ∈ Fin ∧ ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈)) = ∅) → (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
28	20, 22, 26, 27	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
29	18, 28	eqtrid 2783	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
30	16, 29	oveq12d 7428	. . 3 ⊢ (𝜑 → ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))) = (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))))
31		hashcl 14379	. . . . . 6 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
32	3, 4, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
33	32	nn0cnd 12569	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℂ)
34		hashcl 14379	. . . . . 6 ⊢ ((𝑊 ∩ 𝑈) ∈ Fin → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
35	6, 7, 34	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
36	35	nn0cnd 12569	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℂ)
37		hashcl 14379	. . . . . 6 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
38	3, 19, 37	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
39	38	nn0cnd 12569	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℂ)
40		hashcl 14379	. . . . . 6 ⊢ ((𝑊 ∖ 𝑈) ∈ Fin → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
41	6, 21, 40	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
42	41	nn0cnd 12569	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℂ)
43	33, 36, 39, 42	addsub4d 11646	. . 3 ⊢ (𝜑 → (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
44	30, 43	eqtrd 2771	. 2 ⊢ (𝜑 → ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
45		ballotlemgun.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
46		unfi 9190	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑉 ∪ 𝑊) ∈ Fin)
47	3, 6, 46	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑉 ∪ 𝑊) ∈ Fin)
48		ballotth.m	. . . 4 ⊢ 𝑀 ∈ ℕ
49		ballotth.n	. . . 4 ⊢ 𝑁 ∈ ℕ
50		ballotth.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
51		ballotth.p	. . . 4 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
52		ballotth.f	. . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
53		ballotth.e	. . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
54		ballotth.mgtn	. . . 4 ⊢ 𝑁 < 𝑀
55		ballotth.i	. . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
56		ballotth.s	. . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
57		ballotth.r	. . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
58		ballotlemg	. . . 4 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
59	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 34561	. . 3 ⊢ ((𝑈 ∈ Fin ∧ (𝑉 ∪ 𝑊) ∈ Fin) → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
60	45, 47, 59	syl2anc 584	. 2 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
61	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 34561	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
62	45, 3, 61	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
63	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 34561	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑈 ↑ 𝑊) = ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈))))
64	45, 6, 63	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑊) = ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈))))
65	62, 64	oveq12d 7428	. 2 ⊢ (𝜑 → ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
66	44, 60, 65	3eqtr4d 2781	1 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∀wral 3052 {crab 3420 ∖ cdif 3928 ∪ cun 3929 ∩ cin 3930 ∅c0 4313 ifcif 4505 𝒫 cpw 4580 class class class wbr 5124 ↦ cmpt 5206 “ cima 5662 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 Fincfn 8964 infcinf 9458 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 < clt 11274 ≤ cle 11275 − cmin 11471 / cdiv 11899 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ...cfz 13529 ♯chash 14353

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858 df-hash 14354

This theorem is referenced by: ballotlemfrceq 34566

W3C validator