ballotlemfp1 - Mathbox for Thierry Arnoux

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Theorem ballotlemfp1 35003

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 12641	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
9	1, 2, 3, 4, 5, 6, 8	ballotlemfval 35001	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
10	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
11		fzfi 14036	. . . . . . . . . 10 ⊢ (1...(𝐽 − 1)) ∈ Fin
12		inss1 4182	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))
13		ssfi 9167	. . . . . . . . . 10 ⊢ (((1...(𝐽 − 1)) ∈ Fin ∧ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))) → ((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin)
14	11, 12, 13	mp2an 705	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin
15		hashcl 14420	. . . . . . . . 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 12540	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ)
19		diffi 9169	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∈ Fin → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
20	11, 19	ax-mp 5	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
21		hashcl 14420	. . . . . . . . 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 12540	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ
24	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
25		1cnd 11226	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
26	18, 24, 25	subsub4d 11624	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
27		1zzd 12649	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
28	8, 27	zsubcld 12730	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfval 35001	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
30	29	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
31	30	oveq1d 7428	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) − 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1))
32		elnnuz 12927	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
33	7, 32	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
34		fzspl 33260	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘1) → (1...𝐽) = ((1...(𝐽 − 1)) ∪ {𝐽}))
35	34	ineq1d 4165	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶))
36		indir 4232	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))
37	35, 36	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
38	33, 37	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
39	38	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
40		disjsn 4672	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ¬ 𝐽 ∈ 𝐶)
41		incom 4155	. . . . . . . . . . . . 13 ⊢ (𝐶 ∩ {𝐽}) = ({𝐽} ∩ 𝐶)
42	41	eqeq1i 2765	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ({𝐽} ∩ 𝐶) = ∅)
43	40, 42	sylbb1 240	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = ∅)
44	43	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ({𝐽} ∩ 𝐶) = ∅)
45	44	uneq2d 4115	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅))
46		un0 4344	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅) = ((1...(𝐽 − 1)) ∩ 𝐶)
47	45, 46	eqtrdi 2811	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = ((1...(𝐽 − 1)) ∩ 𝐶))
48	39, 47	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = ((1...(𝐽 − 1)) ∩ 𝐶))
49	48	fveq2d 6882	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∩ 𝐶)))
50	34	difeq1d 4073	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶))
51		difundir 4237	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))
52	50, 51	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)))
53	33, 52	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)))
54		disj3 4407	. . . . . . . . . . . 12 ⊢ (({𝐽} ∩ 𝐶) = ∅ ↔ {𝐽} = ({𝐽} ∖ 𝐶))
55	43, 54	sylib 221	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → {𝐽} = ({𝐽} ∖ 𝐶))
56	55	eqcomd 2766	. . . . . . . . . 10 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = {𝐽})
57	56	uneq2d 4115	. . . . . . . . 9 ⊢ (¬ 𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
58	53, 57	sylan9eq 2815	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
59	58	fveq2d 6882	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})))
60	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
61		uzid 12902	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℤ → 𝐽 ∈ (ℤ_≥‘𝐽))
62		uznfz 13665	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘𝐽) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
63	8, 61, 62	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
64	63	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
65		difss 4083	. . . . . . . . . . 11 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))
66	65	sseli 3927	. . . . . . . . . 10 ⊢ (𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶) → 𝐽 ∈ (1...(𝐽 − 1)))
67	64, 66	nsyl 141	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶))
68		ssfi 9167	. . . . . . . . . 10 ⊢ (((1...(𝐽 − 1)) ∈ Fin ∧ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))) → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
69	11, 65, 68	mp2an 705	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
70	67, 69	jctil 529	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶)))
71		hashunsng 14456	. . . . . . . 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 66	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1))
73	59, 72	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1))
74	49, 73	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
75	26, 31, 74	3eqtr4rd 2806	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
76	10, 75	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
77	76	ex 418	. 2 ⊢ (𝜑 → (¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)))
78	9	adantr 486	. . . 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 11226	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
81	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
82	79, 80, 81	addsubd 11614	. . . . 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 6882	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
84	83	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
85		snssi 4746	. . . . . . . . . . 11 ⊢ (𝐽 ∈ 𝐶 → {𝐽} ⊆ 𝐶)
86		dfss2 3917	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 ↔ ({𝐽} ∩ 𝐶) = {𝐽})
87	85, 86	sylib 221	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = {𝐽})
88	87	uneq2d 4115	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽}))
89	88	fveq2d 6882	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
90	89	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
91		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ 𝐶)
92	8	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
93	92, 61, 62	3syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
94	12	sseli 3927	. . . . . . . . . 10 ⊢ (𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶) → 𝐽 ∈ (1...(𝐽 − 1)))
95	93, 94	nsyl 141	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶))
96	95, 14	jctil 529	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶)))
97		hashunsng 14456	. . . . . . . 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 66	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1))
99	84, 90, 98	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1))
100	53	fveq2d 6882	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
101	100	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
102		difin2 4247	. . . . . . . . . . . 12 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ((𝐶 ∖ 𝐶) ∩ {𝐽}))
103		difid 4325	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∖ 𝐶) = ∅
104	103	ineq1i 4162	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = (∅ ∩ {𝐽})
105		0in 4347	. . . . . . . . . . . . 13 ⊢ (∅ ∩ {𝐽}) = ∅
106	104, 105	eqtri 2783	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = ∅
107	102, 106	eqtrdi 2811	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
108	85, 107	syl 18	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
109	108	uneq2d 4115	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅))
110	109	fveq2d 6882	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
111	110	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
112		un0 4344	. . . . . . . . 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 6882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
115	101, 111, 114	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
116	99, 115	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
117	29	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
118	117	oveq1d 7428	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) + 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) + 1))
119	82, 116, 118	3eqtr4d 2805	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
120	78, 119	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
121	120	ex 418	. 2 ⊢ (𝜑 → (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1)))
122	77, 121	jca 521	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3412 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 𝒫 cpw 4557 {csn 4584 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 ℂcc 11122 1c1 11125 + caddc 11127 − cmin 11465 / cdiv 11895 ℕcn 12257 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12887 ...cfz 13561 ♯chash 14394

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-hash 14395

This theorem is used by: ballotlemfc0 35004 ballotlemfcc 35005 ballotlem4 35010 ballotlemi1 35014 ballotlemii 35015 ballotlemic 35018 ballotlem1c 35019

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