ballotlemgun - Mathbox for Thierry Arnoux

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Theorem ballotlemgun 34982

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 4239	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))
2	1	fveq2i 6888	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈)))
3		ballotlemgun.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Fin)
4		infi 9237	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∩ 𝑈) ∈ Fin)
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑉 ∩ 𝑈) ∈ Fin)
6		ballotlemgun.3	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
7		infi 9237	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∩ 𝑈) ∈ Fin)
8	6, 7	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑊 ∩ 𝑈) ∈ Fin)
9		ballotlemgun.4	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ∩ 𝑊) = ∅)
10	9	ineq1d 4172	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∩ 𝑈) = (∅ ∩ 𝑈))
11		inindir 4188	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈))
12		0in 4354	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
13	10, 11, 12	3eqtr3g 2823	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈)) = ∅)
14		hashun 14438	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
16	2, 15	eqtrid 2812	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
17		difundir 4244	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))
18	17	fveq2i 6888	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈)))
19		diffi 9166	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ 𝑈) ∈ Fin)
20	3, 19	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
21		diffi 9166	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∖ 𝑈) ∈ Fin)
22	6, 21	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑊 ∖ 𝑈) ∈ Fin)
23	9	difeq1d 4080	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∖ 𝑈) = (∅ ∖ 𝑈))
24		difindir 4246	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈))
25		0dif 4363	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
26	23, 24, 25	3eqtr3g 2823	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈)) = ∅)
27		hashun 14438	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
29	18, 28	eqtrid 2812	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
30	16, 29	oveq12d 7437	. . 3 ⊢ (𝜑 → ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))) = (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))))
31		hashcl 14412	. . . . . 6 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
32	3, 4, 31	3syl 19	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
33	32	nn0cnd 12584	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℂ)
34		hashcl 14412	. . . . . 6 ⊢ ((𝑊 ∩ 𝑈) ∈ Fin → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
35	6, 7, 34	3syl 19	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
36	35	nn0cnd 12584	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℂ)
37		hashcl 14412	. . . . . 6 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
38	3, 19, 37	3syl 19	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
39	38	nn0cnd 12584	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℂ)
40		hashcl 14412	. . . . . 6 ⊢ ((𝑊 ∖ 𝑈) ∈ Fin → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
41	6, 21, 40	3syl 19	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
42	41	nn0cnd 12584	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℂ)
43	33, 36, 39, 42	addsub4d 11633	. . 3 ⊢ (𝜑 → (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
44	30, 43	eqtrd 2800	. 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 9162	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑉 ∪ 𝑊) ∈ Fin)
47	3, 6, 46	syl2anc 596	. . 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 34981	. . 3 ⊢ ((𝑈 ∈ Fin ∧ (𝑉 ∪ 𝑊) ∈ Fin) → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
60	45, 47, 59	syl2anc 596	. 2 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
61	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 34981	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
62	45, 3, 61	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
63	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 34981	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑈 ↑ 𝑊) = ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈))))
64	45, 6, 63	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑊) = ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈))))
65	62, 64	oveq12d 7437	. 2 ⊢ (𝜑 → ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
66	44, 60, 65	3eqtr4d 2810	1 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ∀wral 3081 {crab 3418 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ∅c0 4286 ifcif 4489 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 “ cima 5666 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 Fincfn 8949 infcinf 9408 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 < clt 11260 ≤ cle 11261 − cmin 11458 / cdiv 11888 ℕcn 12250 ℕ₀cn0 12521 ℤcz 12608 ...cfz 13553 ♯chash 14386

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-n0 12522 df-z 12609 df-uz 12881 df-hash 14387

This theorem is used by: ballotlemfrceq 34986

W3C validator