ballotlemgun - Mathbox for Thierry Arnoux

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Theorem ballotlemgun 29913

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 3834	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))
2	1	fveq2i 6106	. . . . 5 ⊢ (#‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = (#‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈)))
3		ballotlemgun.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Fin)
4		infi 8069	. . . . . . 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 8069	. . . . . . 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 3775	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∩ 𝑈) = (∅ ∩ 𝑈))
11		inindir 3793	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∩ 𝑈) = ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈))
12		0in 3921	. . . . . . 7 ⊢ (∅ ∩ 𝑈) = ∅
13	10, 11, 12	3eqtr3g 2667	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∩ 𝑈) ∩ (𝑊 ∩ 𝑈)) = ∅)
14		hashun 13032	. . . . . 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 1318	. . . . 5 ⊢ (𝜑 → (#‘((𝑉 ∩ 𝑈) ∪ (𝑊 ∩ 𝑈))) = ((#‘(𝑉 ∩ 𝑈)) + (#‘(𝑊 ∩ 𝑈))))
16	2, 15	syl5eq 2656	. . . 4 ⊢ (𝜑 → (#‘((𝑉 ∪ 𝑊) ∩ 𝑈)) = ((#‘(𝑉 ∩ 𝑈)) + (#‘(𝑊 ∩ 𝑈))))
17		difundir 3839	. . . . . 6 ⊢ ((𝑉 ∪ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))
18	17	fveq2i 6106	. . . . 5 ⊢ (#‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = (#‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈)))
19		diffi 8077	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ 𝑈) ∈ Fin)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑉 ∖ 𝑈) ∈ Fin)
21		diffi 8077	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 ∖ 𝑈) ∈ Fin)
22	6, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∖ 𝑈) ∈ Fin)
23	9	difeq1d 3689	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ∩ 𝑊) ∖ 𝑈) = (∅ ∖ 𝑈))
24		difindir 3841	. . . . . . 7 ⊢ ((𝑉 ∩ 𝑊) ∖ 𝑈) = ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈))
25		0dif 3929	. . . . . . 7 ⊢ (∅ ∖ 𝑈) = ∅
26	23, 24, 25	3eqtr3g 2667	. . . . . 6 ⊢ (𝜑 → ((𝑉 ∖ 𝑈) ∩ (𝑊 ∖ 𝑈)) = ∅)
27		hashun 13032	. . . . . 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 1318	. . . . 5 ⊢ (𝜑 → (#‘((𝑉 ∖ 𝑈) ∪ (𝑊 ∖ 𝑈))) = ((#‘(𝑉 ∖ 𝑈)) + (#‘(𝑊 ∖ 𝑈))))
29	18, 28	syl5eq 2656	. . . 4 ⊢ (𝜑 → (#‘((𝑉 ∪ 𝑊) ∖ 𝑈)) = ((#‘(𝑉 ∖ 𝑈)) + (#‘(𝑊 ∖ 𝑈))))
30	16, 29	oveq12d 6567	. . 3 ⊢ (𝜑 → ((#‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (#‘((𝑉 ∪ 𝑊) ∖ 𝑈))) = (((#‘(𝑉 ∩ 𝑈)) + (#‘(𝑊 ∩ 𝑈))) − ((#‘(𝑉 ∖ 𝑈)) + (#‘(𝑊 ∖ 𝑈)))))
31		hashcl 13009	. . . . . 6 ⊢ ((𝑉 ∩ 𝑈) ∈ Fin → (#‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
32	3, 4, 31	3syl 18	. . . . 5 ⊢ (𝜑 → (#‘(𝑉 ∩ 𝑈)) ∈ ℕ₀)
33	32	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘(𝑉 ∩ 𝑈)) ∈ ℂ)
34		hashcl 13009	. . . . . 6 ⊢ ((𝑊 ∩ 𝑈) ∈ Fin → (#‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
35	6, 7, 34	3syl 18	. . . . 5 ⊢ (𝜑 → (#‘(𝑊 ∩ 𝑈)) ∈ ℕ₀)
36	35	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘(𝑊 ∩ 𝑈)) ∈ ℂ)
37		hashcl 13009	. . . . . 6 ⊢ ((𝑉 ∖ 𝑈) ∈ Fin → (#‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
38	3, 19, 37	3syl 18	. . . . 5 ⊢ (𝜑 → (#‘(𝑉 ∖ 𝑈)) ∈ ℕ₀)
39	38	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘(𝑉 ∖ 𝑈)) ∈ ℂ)
40		hashcl 13009	. . . . . 6 ⊢ ((𝑊 ∖ 𝑈) ∈ Fin → (#‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
41	6, 21, 40	3syl 18	. . . . 5 ⊢ (𝜑 → (#‘(𝑊 ∖ 𝑈)) ∈ ℕ₀)
42	41	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘(𝑊 ∖ 𝑈)) ∈ ℂ)
43	33, 36, 39, 42	addsub4d 10318	. . 3 ⊢ (𝜑 → (((#‘(𝑉 ∩ 𝑈)) + (#‘(𝑊 ∩ 𝑈))) − ((#‘(𝑉 ∖ 𝑈)) + (#‘(𝑊 ∖ 𝑈)))) = (((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈))) + ((#‘(𝑊 ∩ 𝑈)) − (#‘(𝑊 ∖ 𝑈)))))
44	30, 43	eqtrd 2644	. 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 8112	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑉 ∪ 𝑊) ∈ Fin)
47	3, 6, 46	syl2anc 691	. . 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 29912	. . 3 ⊢ ((𝑈 ∈ Fin ∧ (𝑉 ∪ 𝑊) ∈ Fin) → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((#‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (#‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
60	45, 47, 59	syl2anc 691	. 2 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((#‘((𝑉 ∪ 𝑊) ∩ 𝑈)) − (#‘((𝑉 ∪ 𝑊) ∖ 𝑈))))
61	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 29912	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈))))
62	45, 3, 61	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑉) = ((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈))))
63	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 29912	. . . 4 ⊢ ((𝑈 ∈ Fin ∧ 𝑊 ∈ Fin) → (𝑈 ↑ 𝑊) = ((#‘(𝑊 ∩ 𝑈)) − (#‘(𝑊 ∖ 𝑈))))
64	45, 6, 63	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑈 ↑ 𝑊) = ((#‘(𝑊 ∩ 𝑈)) − (#‘(𝑊 ∖ 𝑈))))
65	62, 64	oveq12d 6567	. 2 ⊢ (𝜑 → ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)) = (((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈))) + ((#‘(𝑊 ∩ 𝑈)) − (#‘(𝑊 ∖ 𝑈)))))
66	44, 60, 65	3eqtr4d 2654	1 ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-hash 12980

This theorem is referenced by: ballotlemfrceq 29917

W3C validator