ballotlemfp1 - Mathbox for Thierry Arnoux

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Theorem ballotlemfp1 34497

Description: If the 𝐽 th ballot is for A, (𝐹‘𝐶) goes up 1. If the 𝐽 th ballot is for B, (𝐹‘𝐶) goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)

**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)))))
ballotlemfp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑂)
ballotlemfp1.j	⊢ (𝜑 → 𝐽 ∈ ℕ)

**Assertion**
Ref	Expression
ballotlemfp1	⊢ (𝜑 → ((¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)) ∧ (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))))

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐 𝑖,𝑀 𝑖,𝑁 𝑖,𝑂,𝑐 𝐹,𝑐,𝑖 𝐶,𝑖 𝑖,𝐽 𝜑,𝑖 Allowed substitution hints: 𝜑(𝑥,𝑐) 𝐶(𝑥,𝑐) 𝑃(𝑥,𝑖,𝑐) 𝐹(𝑥) 𝐽(𝑥,𝑐) 𝑀(𝑥) 𝑁(𝑥) 𝑂(𝑥)

**Proof of Theorem ballotlemfp1**
Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . 6 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . . 6 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . . . 6 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . . . . 6 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . . . 6 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotlemfp1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑂)
7		ballotlemfp1.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ)
8	7	nnzd 12490	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
9	1, 2, 3, 4, 5, 6, 8	ballotlemfval 34495	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
11		fzfi 13874	. . . . . . . . . 10 ⊢ (1...(𝐽 − 1)) ∈ Fin
12		inss1 4182	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))
13		ssfi 9077	. . . . . . . . . 10 ⊢ (((1...(𝐽 − 1)) ∈ Fin ∧ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))) → ((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin)
14	11, 12, 13	mp2an 692	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin
15		hashcl 14258	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℕ₀)
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℕ₀
17	16	nn0cni 12388	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ)
19		diffi 9079	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∈ Fin → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
20	11, 19	ax-mp 5	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
21		hashcl 14258	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℕ₀)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℕ₀
23	22	nn0cni 12388	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ
24	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
25		1cnd 11102	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
26	18, 24, 25	subsub4d 11498	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
27		1zzd 12498	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
28	8, 27	zsubcld 12577	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfval 34495	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
31	30	oveq1d 7356	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) − 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1))
32		elnnuz 12771	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
33	7, 32	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
34		fzspl 32764	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘1) → (1...𝐽) = ((1...(𝐽 − 1)) ∪ {𝐽}))
35	34	ineq1d 4164	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶))
36		indir 4231	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))
37	35, 36	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
38	33, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
39	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
40		disjsn 4659	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ¬ 𝐽 ∈ 𝐶)
41		incom 4154	. . . . . . . . . . . . 13 ⊢ (𝐶 ∩ {𝐽}) = ({𝐽} ∩ 𝐶)
42	41	eqeq1i 2736	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ({𝐽} ∩ 𝐶) = ∅)
43	40, 42	sylbb1 237	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = ∅)
44	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ({𝐽} ∩ 𝐶) = ∅)
45	44	uneq2d 4113	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅))
46		un0 4339	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅) = ((1...(𝐽 − 1)) ∩ 𝐶)
47	45, 46	eqtrdi 2782	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = ((1...(𝐽 − 1)) ∩ 𝐶))
48	39, 47	eqtrd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = ((1...(𝐽 − 1)) ∩ 𝐶))
49	48	fveq2d 6821	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∩ 𝐶)))
50	34	difeq1d 4070	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶))
51		difundir 4236	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))
52	50, 51	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)))
53	33, 52	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)))
54		disj3 4399	. . . . . . . . . . . 12 ⊢ (({𝐽} ∩ 𝐶) = ∅ ↔ {𝐽} = ({𝐽} ∖ 𝐶))
55	43, 54	sylib 218	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → {𝐽} = ({𝐽} ∖ 𝐶))
56	55	eqcomd 2737	. . . . . . . . . 10 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = {𝐽})
57	56	uneq2d 4113	. . . . . . . . 9 ⊢ (¬ 𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
58	53, 57	sylan9eq 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
59	58	fveq2d 6821	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})))
60	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
61		uzid 12742	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℤ → 𝐽 ∈ (ℤ_≥‘𝐽))
62		uznfz 13505	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘𝐽) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
63	8, 61, 62	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
64	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
65		difss 4081	. . . . . . . . . . 11 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))
66	65	sseli 3925	. . . . . . . . . 10 ⊢ (𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶) → 𝐽 ∈ (1...(𝐽 − 1)))
67	64, 66	nsyl 140	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶))
68		ssfi 9077	. . . . . . . . . 10 ⊢ (((1...(𝐽 − 1)) ∈ Fin ∧ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))) → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
69	11, 65, 68	mp2an 692	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
70	67, 69	jctil 519	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶)))
71		hashunsng 14294	. . . . . . . 8 ⊢ (𝐽 ∈ ℤ → ((((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶)) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
72	60, 70, 71	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1))
73	59, 72	eqtrd 2766	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1))
74	49, 73	oveq12d 7359	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
75	26, 31, 74	3eqtr4rd 2777	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
76	10, 75	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
77	76	ex 412	. 2 ⊢ (𝜑 → (¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)))
78	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
79	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ)
80		1cnd 11102	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
81	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
82	79, 80, 81	addsubd 11488	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) + 1))
83	38	fveq2d 6821	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
84	83	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
85		snssi 4755	. . . . . . . . . . 11 ⊢ (𝐽 ∈ 𝐶 → {𝐽} ⊆ 𝐶)
86		dfss2 3915	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 ↔ ({𝐽} ∩ 𝐶) = {𝐽})
87	85, 86	sylib 218	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = {𝐽})
88	87	uneq2d 4113	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽}))
89	88	fveq2d 6821	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
90	89	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
91		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ 𝐶)
92	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
93	92, 61, 62	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
94	12	sseli 3925	. . . . . . . . . 10 ⊢ (𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶) → 𝐽 ∈ (1...(𝐽 − 1)))
95	93, 94	nsyl 140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶))
96	95, 14	jctil 519	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶)))
97		hashunsng 14294	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → ((((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶)) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1)))
98	91, 96, 97	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1))
99	84, 90, 98	3eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1))
100	53	fveq2d 6821	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
101	100	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
102		difin2 4246	. . . . . . . . . . . 12 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ((𝐶 ∖ 𝐶) ∩ {𝐽}))
103		difid 4321	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∖ 𝐶) = ∅
104	103	ineq1i 4161	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = (∅ ∩ {𝐽})
105		0in 4342	. . . . . . . . . . . . 13 ⊢ (∅ ∩ {𝐽}) = ∅
106	104, 105	eqtri 2754	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = ∅
107	102, 106	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
108	85, 107	syl 17	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
109	108	uneq2d 4113	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅))
110	109	fveq2d 6821	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
111	110	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
112		un0 4339	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅) = ((1...(𝐽 − 1)) ∖ 𝐶)
113	112	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅) = ((1...(𝐽 − 1)) ∖ 𝐶))
114	113	fveq2d 6821	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
115	101, 111, 114	3eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
116	99, 115	oveq12d 7359	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
117	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
118	117	oveq1d 7356	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) + 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) + 1))
119	82, 116, 118	3eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
120	78, 119	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
121	120	ex 412	. 2 ⊢ (𝜑 → (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1)))
122	77, 121	jca 511	1 ⊢ (𝜑 → ((¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)) ∧ (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 {crab 3395 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ⊆ wss 3897 ∅c0 4278 𝒫 cpw 4545 {csn 4571 ↦ cmpt 5167 ‘cfv 6476 (class class class)co 7341 Fincfn 8864 ℂcc 10999 1c1 11002 + caddc 11004 − cmin 11339 / cdiv 11769 ℕcn 12120 ℕ₀cn0 12376 ℤcz 12463 ℤ_≥cuz 12727 ...cfz 13402 ♯chash 14232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-oadd 8384 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-dju 9789 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-hash 14233

This theorem is referenced by: ballotlemfc0 34498 ballotlemfcc 34499 ballotlem4 34504 ballotlemi1 34508 ballotlemii 34509 ballotlemic 34512 ballotlem1c 34513

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