ballotlemgun - Mathbox for Thierry Arnoux

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Theorem ballotlemgun 34516

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 4249	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))
2	1	fveq2i 6861	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = (♯‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈)))
3		ballotlemgun.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Fin)
4		infi 9213	. . . . . . 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 9213	. . . . . . 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 4182	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∩ 𝑈) = (∅ ∩ 𝑈))
11		inindir 4199	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈))
12		0in 4360	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
13	10, 11, 12	3eqtr3g 2787	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈)) = ∅)
14		hashun 14347	. . . . . 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 2776	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = ((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))))
17		difundir 4254	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))
18	17	fveq2i 6861	. . . . 5 ⊢ (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = (♯‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈)))
19		diffi 9139	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ 𝑈) ∈ Fin)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
21		diffi 9139	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∖ 𝑈) ∈ Fin)
22	6, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∖ 𝑈) ∈ Fin)
23	9	difeq1d 4088	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∖ 𝑈) = (∅ ∖ 𝑈))
24		difindir 4256	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈))
25		0dif 4368	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
26	23, 24, 25	3eqtr3g 2787	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈)) = ∅)
27		hashun 14347	. . . . . 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 2776	. . . 4 ⊢ (𝜑 → (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈))))
30	16, 29	oveq12d 7405	. . 3 ⊢ (𝜑 → ((♯‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (♯‘((𝑉 ∪ 𝑊) ∖ 𝑈))) = (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))))
31		hashcl 14321	. . . . . 6 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
32	3, 4, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
33	32	nn0cnd 12505	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∩ 𝑈)) ∈ ℂ)
34		hashcl 14321	. . . . . 6 ⊢ ((𝑊 ∩ 𝑈) ∈ Fin → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
35	6, 7, 34	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
36	35	nn0cnd 12505	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∩ 𝑈)) ∈ ℂ)
37		hashcl 14321	. . . . . 6 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
38	3, 19, 37	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
39	38	nn0cnd 12505	. . . 4 ⊢ (𝜑 → (♯‘(𝑉 ∖ 𝑈)) ∈ ℂ)
40		hashcl 14321	. . . . . 6 ⊢ ((𝑊 ∖ 𝑈) ∈ Fin → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
41	6, 21, 40	3syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
42	41	nn0cnd 12505	. . . 4 ⊢ (𝜑 → (♯‘(𝑊 ∖ 𝑈)) ∈ ℂ)
43	33, 36, 39, 42	addsub4d 11580	. . 3 ⊢ (𝜑 → (((♯‘(𝑉 ∩ 𝑈)) + (♯‘(𝑊 ∩ 𝑈))) − ((♯‘(𝑉 ∖ 𝑈)) + (♯‘(𝑊 ∖ 𝑈)))) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
44	30, 43	eqtrd 2764	. 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 9135	. . . 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 34515	. . 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 34515	. . . 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 34515	. . . 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 7405	. 2 ⊢ (𝜑 → ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)) = (((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) + ((♯‘(𝑊 ∩ 𝑈)) − (♯‘(𝑊 ∖ 𝑈)))))
66	44, 60, 65	3eqtr4d 2774	1 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3405 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ∅c0 4296 ifcif 4488 𝒫 cpw 4563 class class class wbr 5107 ↦ cmpt 5188 “ cima 5641 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 Fincfn 8918 infcinf 9392 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ...cfz 13468 ♯chash 14295

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 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-hash 14296

This theorem is referenced by: ballotlemfrceq 34520

W3C validator