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Theorem hashbc 13803

Description: The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)

**Assertion**
Ref	Expression
hashbc	⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐾

Proof of Theorem hashbc Dummy variables 𝑗 𝑘 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6663	. . . . . 6 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
2	1	oveq1d 7163	. . . . 5 ⊢ (𝑤 = ∅ → ((♯‘𝑤)C𝑘) = ((♯‘∅)C𝑘))
3		pweq 4540	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4	3	rabeqdv 3483	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
5	4	fveq2d 6667	. . . . 5 ⊢ (𝑤 = ∅ → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
6	2, 5	eqeq12d 2835	. . . 4 ⊢ (𝑤 = ∅ → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})))
7	6	ralbidv 3195	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})))
8		fveq2 6663	. . . . . 6 ⊢ (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
9	8	oveq1d 7163	. . . . 5 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤)C𝑘) = ((♯‘𝑦)C𝑘))
10		pweq 4540	. . . . . . 7 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
11	10	rabeqdv 3483	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})
12	11	fveq2d 6667	. . . . 5 ⊢ (𝑤 = 𝑦 → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}))
13	9, 12	eqeq12d 2835	. . . 4 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})))
14	13	ralbidv 3195	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})))
15		fveq2 6663	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘𝑤) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq1d 7163	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((♯‘𝑤)C𝑘) = ((♯‘(𝑦 ∪ {𝑧}))C𝑘))
17		pweq 4540	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
18	17	rabeqdv 3483	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})
19	18	fveq2d 6667	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘}))
20	16, 19	eqeq12d 2835	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
21	20	ralbidv 3195	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
22		fveq2 6663	. . . . . 6 ⊢ (𝑤 = 𝐴 → (♯‘𝑤) = (♯‘𝐴))
23	22	oveq1d 7163	. . . . 5 ⊢ (𝑤 = 𝐴 → ((♯‘𝑤)C𝑘) = ((♯‘𝐴)C𝑘))
24		pweq 4540	. . . . . . 7 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
25	24	rabeqdv 3483	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})
26	25	fveq2d 6667	. . . . 5 ⊢ (𝑤 = 𝐴 → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}))
27	23, 26	eqeq12d 2835	. . . 4 ⊢ (𝑤 = 𝐴 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})))
28	27	ralbidv 3195	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})))
29		hash0 13720	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘∅) = 0)
31		elfz1eq 12910	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
32	30, 31	oveq12d 7166	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (0C0))
33		0nn0 11904	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
34		bcn0 13662	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
36	32, 35	syl6eq 2870	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = 1)
37		pw0 4737	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
38	31	eqcomd 2825	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
39	37	raleqi 3412	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘)
40		0ex 5202	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
41		fveq2 6663	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
42	41, 29	syl6eq 2870	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
43	42	eqeq1d 2821	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((♯‘𝑥) = 𝑘 ↔ 0 = 𝑘))
44	40, 43	ralsn 4611	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
45	39, 44	bitri 277	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
46	38, 45	sylibr 236	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
47		rabid2 3380	. . . . . . . . . . 11 ⊢ (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
48	46, 47	sylibr 236	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
49	37, 48	syl5reqr 2869	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {∅})
50	49	fveq2d 6667	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{∅}))
51		hashsng 13722	. . . . . . . . 9 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
52	40, 51	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∅}) = 1
53	50, 52	syl6eq 2870	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 1)
54	36, 53	eqtr4d 2857	. . . . . 6 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
55	54	adantl 484	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
56	29	oveq1i 7158	. . . . . 6 ⊢ ((♯‘∅)C𝑘) = (0C𝑘)
57		bcval3 13658	. . . . . . . 8 ⊢ ((0 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
58	33, 57	mp3an1 1442	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
59		id 22	. . . . . . . . . . . . . 14 ⊢ (0 = 𝑘 → 0 = 𝑘)
60		0z 11984	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
61		elfz3 12909	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
62	60, 61	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0...0)
63	59, 62	eqeltrrdi 2920	. . . . . . . . . . . . 13 ⊢ (0 = 𝑘 → 𝑘 ∈ (0...0))
64	63	con3i 157	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
65	64	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
66	37	raleqi 3412	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘)
67	43	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
68	40, 67	ralsn 4611	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
69	66, 68	bitri 277	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
70	65, 69	sylibr 236	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
71		rabeq0 4336	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
72	70, 71	sylibr 236	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅)
73	72	fveq2d 6667	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘∅))
74	73, 29	syl6eq 2870	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 0)
75	58, 74	eqtr4d 2857	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
76	56, 75	syl5eq 2866	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
77	55, 76	pm2.61dan 811	. . . 4 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
78	77	rgen 3146	. . 3 ⊢ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
79		oveq2 7156	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((♯‘𝑦)C𝑘) = ((♯‘𝑦)C𝑗))
80		eqeq2 2831	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝑗))
81	80	rabbidv 3479	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗})
82		fveqeq2 6672	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑧) = 𝑗))
83	82	cbvrabv 3490	. . . . . . . 8 ⊢ {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗} = {𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}
84	81, 83	syl6eq 2870	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘} = {𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})
85	84	fveq2d 6667	. . . . . 6 ⊢ (𝑘 = 𝑗 → (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
86	79, 85	eqeq12d 2835	. . . . 5 ⊢ (𝑘 = 𝑗 → (((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})))
87	86	cbvralvw 3448	. . . 4 ⊢ (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
88		simpll 765	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
89		simplr 767	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → ¬ 𝑧 ∈ 𝑦)
90		simprr 771	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
91	83	fveq2i 6666	. . . . . . . . . 10 ⊢ (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})
92	91	eqeq2i 2832	. . . . . . . . 9 ⊢ (((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
93	92	ralbii 3163	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
94	90, 93	sylibr 236	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}))
95		simprl 769	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
96	88, 89, 94, 95	hashbclem 13802	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘}))
97	96	expr 459	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑘 ∈ ℤ) → (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}) → ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
98	97	ralrimdva 3187	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}) → ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
99	87, 98	syl5bi 244	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) → ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
100	7, 14, 21, 28, 78, 99	findcard2s 8751	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}))
101		oveq2 7156	. . . 4 ⊢ (𝑘 = 𝐾 → ((♯‘𝐴)C𝑘) = ((♯‘𝐴)C𝐾))
102		eqeq2 2831	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝐾))
103	102	rabbidv 3479	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
104	103	fveq2d 6667	. . . 4 ⊢ (𝑘 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
105	101, 104	eqeq12d 2835	. . 3 ⊢ (𝑘 = 𝐾 → (((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})))
106	105	rspccva 3620	. 2 ⊢ ((∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
107	100, 106	sylan 582	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 {crab 3140 Vcvv 3493 ∪ cun 3932 ∅c0 4289 𝒫 cpw 4537 {csn 4559 ‘cfv 6348 (class class class)co 7148 Fincfn 8501 0cc0 10529 1c1 10530 ℕ₀cn0 11889 ℤcz 11973 ...cfz 12884 Ccbc 13654 ♯chash 13682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-fz 12885 df-seq 13362 df-fac 13626 df-bc 13655 df-hash 13683

This theorem is referenced by: hashbc2 16334 sylow1lem1 18715 musum 25760 ballotlem1 31737 ballotlem2 31739

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