ballotlemfp1 - Mathbox for Thierry Arnoux

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Theorem ballotlemfp1 34473

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 12638	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
9	1, 2, 3, 4, 5, 6, 8	ballotlemfval 34471	. . . . 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 14010	. . . . . . . . . 10 ⊢ (1...(𝐽 − 1)) ∈ Fin
12		inss1 4245	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∩ 𝐶) ⊆ (1...(𝐽 − 1))
13		ssfi 9212	. . . . . . . . . 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 14392	. . . . . . . . 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 12536	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∩ 𝐶)) ∈ ℂ)
19		diffi 9214	. . . . . . . . . 10 ⊢ ((1...(𝐽 − 1)) ∈ Fin → ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin)
20	11, 19	ax-mp 5	. . . . . . . . 9 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ∈ Fin
21		hashcl 14392	. . . . . . . . 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 12536	. . . . . . 7 ⊢ (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ
24	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
25		1cnd 11254	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
26	18, 24, 25	subsub4d 11649	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
27		1zzd 12646	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
28	8, 27	zsubcld 12725	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
29	1, 2, 3, 4, 5, 6, 28	ballotlemfval 34471	. . . . . . 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 7446	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) − 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) − 1))
32		elnnuz 12920	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
33	7, 32	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
34		fzspl 32798	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (ℤ_≥‘1) → (1...𝐽) = ((1...(𝐽 − 1)) ∪ {𝐽}))
35	34	ineq1d 4227	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∩ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶))
36		indir 4292	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∩ 𝐶) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶))
37	35, 36	eqtrdi 2791	. . . . . . . . . 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 4716	. . . . . . . . . . . 12 ⊢ ((𝐶 ∩ {𝐽}) = ∅ ↔ ¬ 𝐽 ∈ 𝐶)
41		incom 4217	. . . . . . . . . . . . 13 ⊢ (𝐶 ∩ {𝐽}) = ({𝐽} ∩ 𝐶)
42	41	eqeq1i 2740	. . . . . . . . . . . 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 4178	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅))
46		un0 4400	. . . . . . . . 9 ⊢ (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ∅) = ((1...(𝐽 − 1)) ∩ 𝐶)
47	45, 46	eqtrdi 2791	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = ((1...(𝐽 − 1)) ∩ 𝐶))
48	39, 47	eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∩ 𝐶) = ((1...(𝐽 − 1)) ∩ 𝐶))
49	48	fveq2d 6911	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∩ 𝐶)))
50	34	difeq1d 4135	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (ℤ_≥‘1) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶))
51		difundir 4297	. . . . . . . . . . 11 ⊢ (((1...(𝐽 − 1)) ∪ {𝐽}) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶))
52	50, 51	eqtrdi 2791	. . . . . . . . . 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 4460	. . . . . . . . . . . 12 ⊢ (({𝐽} ∩ 𝐶) = ∅ ↔ {𝐽} = ({𝐽} ∖ 𝐶))
55	43, 54	sylib 218	. . . . . . . . . . 11 ⊢ (¬ 𝐽 ∈ 𝐶 → {𝐽} = ({𝐽} ∖ 𝐶))
56	55	eqcomd 2741	. . . . . . . . . 10 ⊢ (¬ 𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = {𝐽})
57	56	uneq2d 4178	. . . . . . . . 9 ⊢ (¬ 𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
58	53, 57	sylan9eq 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((1...𝐽) ∖ 𝐶) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽}))
59	58	fveq2d 6911	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ {𝐽})))
60	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → 𝐽 ∈ ℤ)
61		uzid 12891	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℤ → 𝐽 ∈ (ℤ_≥‘𝐽))
62		uznfz 13647	. . . . . . . . . . . 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 4146	. . . . . . . . . . 11 ⊢ ((1...(𝐽 − 1)) ∖ 𝐶) ⊆ (1...(𝐽 − 1))
66	65	sseli 3991	. . . . . . . . . 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 9212	. . . . . . . . . 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 14428	. . . . . . . 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 2775	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1))
74	49, 73	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − ((♯‘((1...(𝐽 − 1)) ∖ 𝐶)) + 1)))
75	26, 31, 74	3eqtr4rd 2786	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1))
76	10, 75	eqtrd 2775	. . 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 11254	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → 1 ∈ ℂ)
81	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...(𝐽 − 1)) ∖ 𝐶)) ∈ ℂ)
82	79, 80, 81	addsubd 11639	. . . . 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 6911	. . . . . . . 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 4813	. . . . . . . . . . 11 ⊢ (𝐽 ∈ 𝐶 → {𝐽} ⊆ 𝐶)
86		dfss2 3981	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 ↔ ({𝐽} ∩ 𝐶) = {𝐽})
87	85, 86	sylib 218	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∩ 𝐶) = {𝐽})
88	87	uneq2d 4178	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∩ 𝐶) ∪ ({𝐽} ∩ 𝐶)) = (((1...(𝐽 − 1)) ∩ 𝐶) ∪ {𝐽}))
89	88	fveq2d 6911	. . . . . . . 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 3991	. . . . . . . . . 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 14428	. . . . . . . 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 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∩ 𝐶)) = ((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) + 1))
100	53	fveq2d 6911	. . . . . . . 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 4307	. . . . . . . . . . . 12 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ((𝐶 ∖ 𝐶) ∩ {𝐽}))
103		difid 4382	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∖ 𝐶) = ∅
104	103	ineq1i 4224	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = (∅ ∩ {𝐽})
105		0in 4403	. . . . . . . . . . . . 13 ⊢ (∅ ∩ {𝐽}) = ∅
106	104, 105	eqtri 2763	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ 𝐶) ∩ {𝐽}) = ∅
107	102, 106	eqtrdi 2791	. . . . . . . . . . 11 ⊢ ({𝐽} ⊆ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
108	85, 107	syl 17	. . . . . . . . . 10 ⊢ (𝐽 ∈ 𝐶 → ({𝐽} ∖ 𝐶) = ∅)
109	108	uneq2d 4178	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝐶 → (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ({𝐽} ∖ 𝐶)) = (((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅))
110	109	fveq2d 6911	. . . . . . . 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 4400	. . . . . . . . 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 6911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘(((1...(𝐽 − 1)) ∖ 𝐶) ∪ ∅)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
115	101, 111, 114	3eqtrd 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (♯‘((1...𝐽) ∖ 𝐶)) = (♯‘((1...(𝐽 − 1)) ∖ 𝐶)))
116	99, 115	oveq12d 7449	. . . . 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 7446	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → (((𝐹‘𝐶)‘(𝐽 − 1)) + 1) = (((♯‘((1...(𝐽 − 1)) ∩ 𝐶)) − (♯‘((1...(𝐽 − 1)) ∖ 𝐶))) + 1))
119	82, 116, 118	3eqtr4d 2785	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐶) → ((♯‘((1...𝐽) ∩ 𝐶)) − (♯‘((1...𝐽) ∖ 𝐶))) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))
120	78, 119	eqtrd 2775	. . 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 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 1c1 11154 + caddc 11156 − cmin 11490 / cdiv 11918 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ♯chash 14366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367

This theorem is referenced by: ballotlemfc0 34474 ballotlemfcc 34475 ballotlem4 34480 ballotlemi1 34484 ballotlemii 34485 ballotlemic 34488 ballotlem1c 34489

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