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Theorem hashbc 13551

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 6446	. . . . . 6 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
2	1	oveq1d 6937	. . . . 5 ⊢ (𝑤 = ∅ → ((♯‘𝑤)C𝑘) = ((♯‘∅)C𝑘))
3		pweq 4381	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4	3	rabeqdv 3390	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
5	4	fveq2d 6450	. . . . 5 ⊢ (𝑤 = ∅ → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
6	2, 5	eqeq12d 2792	. . . 4 ⊢ (𝑤 = ∅ → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})))
7	6	ralbidv 3167	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})))
8		fveq2 6446	. . . . . 6 ⊢ (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
9	8	oveq1d 6937	. . . . 5 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤)C𝑘) = ((♯‘𝑦)C𝑘))
10		pweq 4381	. . . . . . 7 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
11	10	rabeqdv 3390	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})
12	11	fveq2d 6450	. . . . 5 ⊢ (𝑤 = 𝑦 → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}))
13	9, 12	eqeq12d 2792	. . . 4 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})))
14	13	ralbidv 3167	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})))
15		fveq2 6446	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘𝑤) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq1d 6937	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((♯‘𝑤)C𝑘) = ((♯‘(𝑦 ∪ {𝑧}))C𝑘))
17		pweq 4381	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
18	17	rabeqdv 3390	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})
19	18	fveq2d 6450	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘}))
20	16, 19	eqeq12d 2792	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
21	20	ralbidv 3167	. . 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 6446	. . . . . 6 ⊢ (𝑤 = 𝐴 → (♯‘𝑤) = (♯‘𝐴))
23	22	oveq1d 6937	. . . . 5 ⊢ (𝑤 = 𝐴 → ((♯‘𝑤)C𝑘) = ((♯‘𝐴)C𝑘))
24		pweq 4381	. . . . . . 7 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
25	24	rabeqdv 3390	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})
26	25	fveq2d 6450	. . . . 5 ⊢ (𝑤 = 𝐴 → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}))
27	23, 26	eqeq12d 2792	. . . 4 ⊢ (𝑤 = 𝐴 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})))
28	27	ralbidv 3167	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})))
29		hash0 13473	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘∅) = 0)
31		elfz1eq 12669	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
32	30, 31	oveq12d 6940	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (0C0))
33		0nn0 11659	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
34		bcn0 13415	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
36	32, 35	syl6eq 2829	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = 1)
37		pw0 4574	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
38	31	eqcomd 2783	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
39	37	raleqi 3337	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘)
40		0ex 5026	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
41		fveq2 6446	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
42	41, 29	syl6eq 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
43	42	eqeq1d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((♯‘𝑥) = 𝑘 ↔ 0 = 𝑘))
44	40, 43	ralsn 4449	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
45	39, 44	bitri 267	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
46	38, 45	sylibr 226	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
47		rabid2 3304	. . . . . . . . . . 11 ⊢ (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
48	46, 47	sylibr 226	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
49	37, 48	syl5reqr 2828	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {∅})
50	49	fveq2d 6450	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{∅}))
51		hashsng 13474	. . . . . . . . 9 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
52	40, 51	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∅}) = 1
53	50, 52	syl6eq 2829	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 1)
54	36, 53	eqtr4d 2816	. . . . . 6 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
55	54	adantl 475	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
56	29	oveq1i 6932	. . . . . 6 ⊢ ((♯‘∅)C𝑘) = (0C𝑘)
57		bcval3 13411	. . . . . . . 8 ⊢ ((0 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
58	33, 57	mp3an1 1521	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
59		id 22	. . . . . . . . . . . . . 14 ⊢ (0 = 𝑘 → 0 = 𝑘)
60		0z 11739	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
61		elfz3 12668	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
62	60, 61	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0...0)
63	59, 62	syl6eqelr 2867	. . . . . . . . . . . . 13 ⊢ (0 = 𝑘 → 𝑘 ∈ (0...0))
64	63	con3i 152	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
65	64	adantl 475	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
66	37	raleqi 3337	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘)
67	43	notbid 310	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
68	40, 67	ralsn 4449	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
69	66, 68	bitri 267	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
70	65, 69	sylibr 226	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
71		rabeq0 4186	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
72	70, 71	sylibr 226	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅)
73	72	fveq2d 6450	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘∅))
74	73, 29	syl6eq 2829	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 0)
75	58, 74	eqtr4d 2816	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
76	56, 75	syl5eq 2825	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
77	55, 76	pm2.61dan 803	. . . 4 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
78	77	rgen 3103	. . 3 ⊢ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
79		oveq2 6930	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((♯‘𝑦)C𝑘) = ((♯‘𝑦)C𝑗))
80		eqeq2 2788	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝑗))
81	80	rabbidv 3385	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗})
82		fveqeq2 6455	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑧) = 𝑗))
83	82	cbvrabv 3395	. . . . . . . 8 ⊢ {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗} = {𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}
84	81, 83	syl6eq 2829	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘} = {𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})
85	84	fveq2d 6450	. . . . . 6 ⊢ (𝑘 = 𝑗 → (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
86	79, 85	eqeq12d 2792	. . . . 5 ⊢ (𝑘 = 𝑗 → (((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})))
87	86	cbvralv 3366	. . . 4 ⊢ (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
88		simpll 757	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
89		simplr 759	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → ¬ 𝑧 ∈ 𝑦)
90		simprr 763	. . . . . . . 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 6449	. . . . . . . . . 10 ⊢ (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})
92	91	eqeq2i 2789	. . . . . . . . 9 ⊢ (((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
93	92	ralbii 3161	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
94	90, 93	sylibr 226	. . . . . . 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 761	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
96	88, 89, 94, 95	hashbclem 13550	. . . . . 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 450	. . . . 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 3150	. . . 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 234	. . 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 8489	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}))
101		oveq2 6930	. . . 4 ⊢ (𝑘 = 𝐾 → ((♯‘𝐴)C𝑘) = ((♯‘𝐴)C𝐾))
102		eqeq2 2788	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝐾))
103	102	rabbidv 3385	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
104	103	fveq2d 6450	. . . 4 ⊢ (𝑘 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
105	101, 104	eqeq12d 2792	. . 3 ⊢ (𝑘 = 𝐾 → (((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})))
106	105	rspccva 3509	. 2 ⊢ ((∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
107	100, 106	sylan 575	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 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 {crab 3093 Vcvv 3397 ∪ cun 3789 ∅c0 4140 𝒫 cpw 4378 {csn 4397 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 0cc0 10272 1c1 10273 ℕ₀cn0 11642 ℤcz 11728 ...cfz 12643 Ccbc 13407 ♯chash 13435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-seq 13120 df-fac 13379 df-bc 13408 df-hash 13436

This theorem is referenced by: hashbc2 16114 sylow1lem1 18397 musum 25369 ballotlem1 31147 ballotlem2 31149

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