ballotlemfp1 - Mathbox for Thierry Arnoux

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Theorem ballotlemfp1 32142

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 12264	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
9	1, 2, 3, 4, 5, 6, 8	ballotlemfval 32140	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
10	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘𝐽) = ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))))
11		fzfi 13528	. . . . . . . . . 10 ⊢ (1...(𝐽 − 1)) ∈ Fin
12		inss1 4133	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))
13		ssfi 8840	. . . . . . . . . 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 13906	. . . . . . . . 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 12085	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ)
19		diffi 8895	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∈ Fin → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
20	11, 19	ax-mp 5	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
21		hashcl 13906	. . . . . . . . 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 12085	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ
24	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
25		1cnd 10811	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
26	18, 24, 25	subsub4d 11203	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
27		1zzd 12191	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
28	8, 27	zsubcld 12270	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfval 32140	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
30	29	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
31	30	oveq1d 7217	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) − 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1))
32		elnnuz 12461	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
33	7, 32	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
34		fzspl 30803	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘1) → (1...𝐽) = ((1...(𝐽 − 1)) ∪ {𝐽}))
35	34	ineq1d 4116	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶))
36		indir 4180	. . . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)))
40		disjsn 4617	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ¬ 𝐽 ∈ 𝐶)
41		incom 4105	. . . . . . . . . . . . 13 ⊢ (𝐶 ∩ {𝐽}) = ({𝐽} ∩ 𝐶)
42	41	eqeq1i 2739	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ({𝐽} ∩ 𝐶) = ∅)
43	40, 42	sylbb1 240	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = ∅)
44	43	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ({𝐽} ∩ 𝐶) = ∅)
45	44	uneq2d 4067	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅))
46		un0 4295	. . . . . . . . 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 6710	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∩ 𝐶)))
50	34	difeq1d 4026	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶))
51		difundir 4185	. . . . . . . . . . 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 4358	. . . . . . . . . . . 12 ⊢ (({𝐽} ∩ 𝐶) = ∅ ↔ {𝐽} = ({𝐽} ∖ 𝐶))
55	43, 54	sylib 221	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → {𝐽} = ({𝐽} ∖ 𝐶))
56	55	eqcomd 2740	. . . . . . . . . 10 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = {𝐽})
57	56	uneq2d 4067	. . . . . . . . 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 6710	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})))
60	8	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
61		uzid 12436	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℤ → 𝐽 ∈ (ℤ_≥‘𝐽))
62		uznfz 13178	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘𝐽) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
63	8, 61, 62	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
64	63	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
65		difss 4036	. . . . . . . . . . 11 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))
66	65	sseli 3887	. . . . . . . . . 10 ⊢ (𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶) → 𝐽 ∈ (1...(𝐽 − 1)))
67	64, 66	nsyl 142	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶))
68		ssfi 8840	. . . . . . . . . 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 523	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∖ 𝐶)))
71		hashunsng 13942	. . . . . . . 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 7220	. . . . 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 416	. 2 ⊢ (𝜑 → (¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)))
78	9	adantr 484	. . . 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 10811	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
81	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
82	79, 80, 81	addsubd 11193	. . . . 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 6710	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
84	83	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))))
85		snssi 4711	. . . . . . . . . . 11 ⊢ (𝐽 ∈ 𝐶 → {𝐽} ⊆ 𝐶)
86		df-ss 3874	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 ↔ ({𝐽} ∩ 𝐶) = {𝐽})
87	85, 86	sylib 221	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = {𝐽})
88	87	uneq2d 4067	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽}))
89	88	fveq2d 6710	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
90	89	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽})))
91		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ 𝐶)
92	8	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
93	92, 61, 62	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ (1...(𝐽 − 1)))
94	12	sseli 3887	. . . . . . . . . 10 ⊢ (𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶) → 𝐽 ∈ (1...(𝐽 − 1)))
95	93, 94	nsyl 142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶))
96	95, 14	jctil 523	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∈ Fin ∧ ¬ 𝐽 ∈ ((1...(𝐽 − 1)) ∩ 𝐶)))
97		hashunsng 13942	. . . . . . . 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 6710	. . . . . . . 8 ⊢ (𝜑 → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
101	100	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))))
102		difin2 4196	. . . . . . . . . . . 12 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ((𝐶 ∖ 𝐶) ∩ {𝐽}))
103		difid 4275	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∖ 𝐶) = ∅
104	103	ineq1i 4113	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = (∅ ∩ {𝐽})
105		0in 4298	. . . . . . . . . . . . 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 4067	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅))
110	109	fveq2d 6710	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐶 → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
111	110	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)))
112		un0 4295	. . . . . . . . 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 6710	. . . . . . 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 7220	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
117	29	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐽 − 1)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))))
118	117	oveq1d 7217	. . . . 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 416	. 2 ⊢ (𝜑 → (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1)))
122	77, 121	jca 515	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 399 = wceq 1543 ∈ wcel 2110 {crab 3058 ∖ cdif 3854 ∪ cun 3855 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 {csn 4531 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 ℂcc 10710 1c1 10713 + caddc 10715 − cmin 11045 / cdiv 11472 ℕcn 11813 ℕ₀cn0 12073 ℤcz 12159 ℤ_≥cuz 12421 ...cfz 13078 ♯chash 13879

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-hash 13880

This theorem is referenced by: ballotlemfc0 32143 ballotlemfcc 32144 ballotlem4 32149 ballotlemi1 32153 ballotlemii 32154 ballotlemic 32157 ballotlem1c 32158

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