ballotlemgun - Mathbox for Thierry Arnoux

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Theorem ballotlemgun 32391

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 4206	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))
2	1	fveq2i 6759	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈)))
3		ballotlemgun.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Fin)
4		infi 8972	. . . . . . 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 8972	. . . . . . 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 4142	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∩ 𝑈) = (∅ ∩ 𝑈))
11		inindir 4158	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈))
12		0in 4324	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
13	10, 11, 12	3eqtr3g 2802	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈)) = ∅)
14		hashun 14025	. . . . . 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 1369	. . . . 5 ⊢ (𝜑 → (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
16	2, 15	syl5eq 2791	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
17		difundir 4211	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))
18	17	fveq2i 6759	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈)))
19		diffi 8979	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ 𝑈) ∈ Fin)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
21		diffi 8979	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∖ 𝑈) ∈ Fin)
22	6, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∖ 𝑈) ∈ Fin)
23	9	difeq1d 4052	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∖ 𝑈) = (∅ ∖ 𝑈))
24		difindir 4213	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈))
25		0dif 4332	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
26	23, 24, 25	3eqtr3g 2802	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈)) = ∅)
27		hashun 14025	. . . . . 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 1369	. . . . 5 ⊢ (𝜑 → (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
29	18, 28	syl5eq 2791	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
30	16, 29	oveq12d 7273	. . 3 ⊢ (𝜑 → ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))) = (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))))
31		hashcl 13999	. . . . . 6 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
32	3, 4, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
33	32	nn0cnd 12225	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℂ)
34		hashcl 13999	. . . . . 6 ⊢ ((𝑊 ∩ 𝑈) ∈ Fin → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
35	6, 7, 34	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
36	35	nn0cnd 12225	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℂ)
37		hashcl 13999	. . . . . 6 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
38	3, 19, 37	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
39	38	nn0cnd 12225	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℂ)
40		hashcl 13999	. . . . . 6 ⊢ ((𝑊 ∖ 𝑈) ∈ Fin → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
41	6, 21, 40	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
42	41	nn0cnd 12225	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℂ)
43	33, 36, 39, 42	addsub4d 11309	. . 3 ⊢ (𝜑 → (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
44	30, 43	eqtrd 2778	. 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 8917	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑉 ∪ 𝑊) ∈ Fin)
47	3, 6, 46	syl2anc 583	. . 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 32390	. . 3 ⊢ ((𝑈 ∈ Fin ∧ (𝑉 ∪ 𝑊) ∈ Fin) → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
60	45, 47, 59	syl2anc 583	. 2 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
61	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 32390	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
62	45, 3, 61	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
63	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 32390	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑈 ↑ 𝑊) = ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈))))
64	45, 6, 63	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑊) = ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈))))
65	62, 64	oveq12d 7273	. 2 ⊢ (𝜑 → ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
66	44, 60, 65	3eqtr4d 2788	1 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ∅c0 4253 ifcif 4456 𝒫 cpw 4530 class class class wbr 5070 ↦ cmpt 5153 “ cima 5583 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 Fincfn 8691 infcinf 9130 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ...cfz 13168 ♯chash 13972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-hash 13973

This theorem is referenced by: ballotlemfrceq 32395

W3C validator