ballotlemfp1 - Mathbox for Thierry Arnoux

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Theorem ballotlemfp1 34676

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 12541	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
9	1, 2, 3, 4, 5, 6, 8	ballotlemfval 34674	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
11		fzfi 13925	. . . . . . . . . 10 ⊢ (1...(𝐽 − 1)) ∈ Fin
12		inss1 4165	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))
13		ssfi 9097	. . . . . . . . . 10 ⊢ (((1...(𝐽 − 1)) ∈ Fin ∧ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))) → ((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin)
14	11, 12, 13	mp2an 698	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin
15		hashcl 14309	. . . . . . . . 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 12440	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ)
19		diffi 9099	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∈ Fin → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
20	11, 19	ax-mp 5	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
21		hashcl 14309	. . . . . . . . 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 12440	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ
24	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
25		1cnd 11130	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
26	18, 24, 25	subsub4d 11527	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
27		1zzd 12549	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
28	8, 27	zsubcld 12629	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfval 34674	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
30	29	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
31	30	oveq1d 7371	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) − 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1))
32		elnnuz 12819	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
33	7, 32	sylib 219	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
34		fzspl 32881	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘1) → (1...𝐽) = ((1...(𝐽 − 1)) ∪ {𝐽}))
35	34	ineq1d 4148	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶))
36		indir 4214	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))
37	35, 36	eqtrdi 2790	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
40		disjsn 4643	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ¬ 𝐽 ∈ 𝐶)
41		incom 4138	. . . . . . . . . . . . 13 ⊢ (𝐶 ∩ {𝐽}) = ({𝐽} ∩ 𝐶)
42	41	eqeq1i 2744	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ({𝐽} ∩ 𝐶) = ∅)
43	40, 42	sylbb1 238	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = ∅)
44	43	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ({𝐽} ∩ 𝐶) = ∅)
45	44	uneq2d 4098	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅))
46		un0 4322	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅) = ((1...(𝐽 − 1)) ∩ 𝐶)
47	45, 46	eqtrdi 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = ((1...(𝐽 − 1)) ∩ 𝐶))
48	39, 47	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = ((1...(𝐽 − 1)) ∩ 𝐶))
49	48	fveq2d 6831	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∩ 𝐶)))
50	34	difeq1d 4056	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶))
51		difundir 4219	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))
52	50, 51	eqtrdi 2790	. . . . . . . . . 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 4382	. . . . . . . . . . . 12 ⊢ (({𝐽} ∩ 𝐶) = ∅ ↔ {𝐽} = ({𝐽} ∖ 𝐶))
55	43, 54	sylib 219	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → {𝐽} = ({𝐽} ∖ 𝐶))
56	55	eqcomd 2745	. . . . . . . . . 10 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = {𝐽})
57	56	uneq2d 4098	. . . . . . . . 9 ⊢ (¬ 𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
58	53, 57	sylan9eq 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
59	58	fveq2d 6831	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})))
60	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
61		uzid 12794	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℤ → 𝐽 ∈ (ℤ_≥‘𝐽))
62		uznfz 13555	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘𝐽) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
63	8, 61, 62	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
64	63	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
65		difss 4066	. . . . . . . . . . 11 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))
66	65	sseli 3911	. . . . . . . . . 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 9097	. . . . . . . . . 10 ⊢ (((1...(𝐽 − 1)) ∈ Fin ∧ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))) → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
69	11, 65, 68	mp2an 698	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
70	67, 69	jctil 524	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶)))
71		hashunsng 14345	. . . . . . . 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 2774	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1))
74	49, 73	oveq12d 7374	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
75	26, 31, 74	3eqtr4rd 2785	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
76	10, 75	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
77	76	ex 413	. 2 ⊢ (𝜑 → (¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)))
78	9	adantr 481	. . . 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 11130	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
81	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
82	79, 80, 81	addsubd 11517	. . . . 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 6831	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
84	83	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
85		snssi 4717	. . . . . . . . . . 11 ⊢ (𝐽 ∈ 𝐶 → {𝐽} ⊆ 𝐶)
86		dfss2 3901	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 ↔ ({𝐽} ∩ 𝐶) = {𝐽})
87	85, 86	sylib 219	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = {𝐽})
88	87	uneq2d 4098	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽}))
89	88	fveq2d 6831	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
90	89	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
91		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ 𝐶)
92	8	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
93	92, 61, 62	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
94	12	sseli 3911	. . . . . . . . . 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 524	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶)))
97		hashunsng 14345	. . . . . . . 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 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1))
100	53	fveq2d 6831	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
101	100	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
102		difin2 4229	. . . . . . . . . . . 12 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ((𝐶 ∖ 𝐶) ∩ {𝐽}))
103		difid 4304	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∖ 𝐶) = ∅
104	103	ineq1i 4145	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = (∅ ∩ {𝐽})
105		0in 4325	. . . . . . . . . . . . 13 ⊢ (∅ ∩ {𝐽}) = ∅
106	104, 105	eqtri 2762	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = ∅
107	102, 106	eqtrdi 2790	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
108	85, 107	syl 17	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
109	108	uneq2d 4098	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅))
110	109	fveq2d 6831	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
111	110	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
112		un0 4322	. . . . . . . . 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 6831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
115	101, 111, 114	3eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
116	99, 115	oveq12d 7374	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
117	29	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
118	117	oveq1d 7371	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) + 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) + 1))
119	82, 116, 118	3eqtr4d 2784	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
120	78, 119	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
121	120	ex 413	. 2 ⊢ (𝜑 → (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1)))
122	77, 121	jca 516	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 396 = wceq 1547 ∈ wcel 2119 {crab 3391 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 𝒫 cpw 4529 {csn 4555 ↦ cmpt 5153 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℂcc 11027 1c1 11030 + caddc 11032 − cmin 11368 / cdiv 11798 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ...cfz 13452 ♯chash 14283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-hash 14284

This theorem is referenced by: ballotlemfc0 34677 ballotlemfcc 34678 ballotlem4 34683 ballotlemi1 34687 ballotlemii 34688 ballotlemic 34691 ballotlem1c 34692

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