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Theorem hashbc 13094
 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.
StepHypRef Expression
1 fveq2 6103 . . . . . 6 (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
21oveq1d 6564 . . . . 5 (𝑤 = ∅ → ((#‘𝑤)C𝑘) = ((#‘∅)C𝑘))
3 pweq 4111 . . . . . . 7 (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4 rabeq 3166 . . . . . . 7 (𝒫 𝑤 = 𝒫 ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
53, 4syl 17 . . . . . 6 (𝑤 = ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
65fveq2d 6107 . . . . 5 (𝑤 = ∅ → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
72, 6eqeq12d 2625 . . . 4 (𝑤 = ∅ → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})))
87ralbidv 2969 . . 3 (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})))
9 fveq2 6103 . . . . . 6 (𝑤 = 𝑦 → (#‘𝑤) = (#‘𝑦))
109oveq1d 6564 . . . . 5 (𝑤 = 𝑦 → ((#‘𝑤)C𝑘) = ((#‘𝑦)C𝑘))
11 pweq 4111 . . . . . . 7 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
12 rabeq 3166 . . . . . . 7 (𝒫 𝑤 = 𝒫 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})
1311, 12syl 17 . . . . . 6 (𝑤 = 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})
1413fveq2d 6107 . . . . 5 (𝑤 = 𝑦 → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}))
1510, 14eqeq12d 2625 . . . 4 (𝑤 = 𝑦 → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})))
1615ralbidv 2969 . . 3 (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})))
17 fveq2 6103 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (#‘𝑤) = (#‘(𝑦 ∪ {𝑧})))
1817oveq1d 6564 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → ((#‘𝑤)C𝑘) = ((#‘(𝑦 ∪ {𝑧}))C𝑘))
19 pweq 4111 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
20 rabeq 3166 . . . . . . 7 (𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})
2119, 20syl 17 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})
2221fveq2d 6107 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘}))
2318, 22eqeq12d 2625 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
2423ralbidv 2969 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
25 fveq2 6103 . . . . . 6 (𝑤 = 𝐴 → (#‘𝑤) = (#‘𝐴))
2625oveq1d 6564 . . . . 5 (𝑤 = 𝐴 → ((#‘𝑤)C𝑘) = ((#‘𝐴)C𝑘))
27 pweq 4111 . . . . . . 7 (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
28 rabeq 3166 . . . . . . 7 (𝒫 𝑤 = 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})
2927, 28syl 17 . . . . . 6 (𝑤 = 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})
3029fveq2d 6107 . . . . 5 (𝑤 = 𝐴 → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}))
3126, 30eqeq12d 2625 . . . 4 (𝑤 = 𝐴 → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})))
3231ralbidv 2969 . . 3 (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})))
33 hash0 13019 . . . . . . . . . 10 (#‘∅) = 0
3433a1i 11 . . . . . . . . 9 (𝑘 ∈ (0...0) → (#‘∅) = 0)
35 elfz1eq 12223 . . . . . . . . 9 (𝑘 ∈ (0...0) → 𝑘 = 0)
3634, 35oveq12d 6567 . . . . . . . 8 (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = (0C0))
37 0nn0 11184 . . . . . . . . 9 0 ∈ ℕ0
38 bcn0 12959 . . . . . . . . 9 (0 ∈ ℕ0 → (0C0) = 1)
3937, 38ax-mp 5 . . . . . . . 8 (0C0) = 1
4036, 39syl6eq 2660 . . . . . . 7 (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = 1)
41 pw0 4283 . . . . . . . . . 10 𝒫 ∅ = {∅}
4235eqcomd 2616 . . . . . . . . . . . 12 (𝑘 ∈ (0...0) → 0 = 𝑘)
4341raleqi 3119 . . . . . . . . . . . . 13 (∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (#‘𝑥) = 𝑘)
44 0ex 4718 . . . . . . . . . . . . . 14 ∅ ∈ V
45 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
4645, 33syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (#‘𝑥) = 0)
4746eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((#‘𝑥) = 𝑘 ↔ 0 = 𝑘))
4844, 47ralsn 4169 . . . . . . . . . . . . 13 (∀𝑥 ∈ {∅} (#‘𝑥) = 𝑘 ↔ 0 = 𝑘)
4943, 48bitri 263 . . . . . . . . . . . 12 (∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘 ↔ 0 = 𝑘)
5042, 49sylibr 223 . . . . . . . . . . 11 (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘)
51 rabid2 3096 . . . . . . . . . . 11 (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘)
5250, 51sylibr 223 . . . . . . . . . 10 (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
5341, 52syl5reqr 2659 . . . . . . . . 9 (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = {∅})
5453fveq2d 6107 . . . . . . . 8 (𝑘 ∈ (0...0) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = (#‘{∅}))
55 hashsng 13020 . . . . . . . . 9 (∅ ∈ V → (#‘{∅}) = 1)
5644, 55ax-mp 5 . . . . . . . 8 (#‘{∅}) = 1
5754, 56syl6eq 2660 . . . . . . 7 (𝑘 ∈ (0...0) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = 1)
5840, 57eqtr4d 2647 . . . . . 6 (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
5958adantl 481 . . . . 5 ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
6033oveq1i 6559 . . . . . 6 ((#‘∅)C𝑘) = (0C𝑘)
61 bcval3 12955 . . . . . . . 8 ((0 ∈ ℕ0𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
6237, 61mp3an1 1403 . . . . . . 7 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
63 id 22 . . . . . . . . . . . . . 14 (0 = 𝑘 → 0 = 𝑘)
64 0z 11265 . . . . . . . . . . . . . . 15 0 ∈ ℤ
65 elfz3 12222 . . . . . . . . . . . . . . 15 (0 ∈ ℤ → 0 ∈ (0...0))
6664, 65ax-mp 5 . . . . . . . . . . . . . 14 0 ∈ (0...0)
6763, 66syl6eqelr 2697 . . . . . . . . . . . . 13 (0 = 𝑘𝑘 ∈ (0...0))
6867con3i 149 . . . . . . . . . . . 12 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
6968adantl 481 . . . . . . . . . . 11 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
7041raleqi 3119 . . . . . . . . . . . 12 (∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (#‘𝑥) = 𝑘)
7147notbid 307 . . . . . . . . . . . . 13 (𝑥 = ∅ → (¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
7244, 71ralsn 4169 . . . . . . . . . . . 12 (∀𝑥 ∈ {∅} ¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
7370, 72bitri 263 . . . . . . . . . . 11 (∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
7469, 73sylibr 223 . . . . . . . . . 10 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘)
75 rabeq0 3911 . . . . . . . . . 10 ({𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘)
7674, 75sylibr 223 . . . . . . . . 9 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = ∅)
7776fveq2d 6107 . . . . . . . 8 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = (#‘∅))
7877, 33syl6eq 2660 . . . . . . 7 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = 0)
7962, 78eqtr4d 2647 . . . . . 6 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
8060, 79syl5eq 2656 . . . . 5 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
8159, 80pm2.61dan 828 . . . 4 (𝑘 ∈ ℤ → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
8281rgen 2906 . . 3 𝑘 ∈ ℤ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
83 oveq2 6557 . . . . . 6 (𝑘 = 𝑗 → ((#‘𝑦)C𝑘) = ((#‘𝑦)C𝑗))
84 eqeq2 2621 . . . . . . . . 9 (𝑘 = 𝑗 → ((#‘𝑥) = 𝑘 ↔ (#‘𝑥) = 𝑗))
8584rabbidv 3164 . . . . . . . 8 (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗})
86 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝑧 → (#‘𝑥) = (#‘𝑧))
8786eqeq1d 2612 . . . . . . . . 9 (𝑥 = 𝑧 → ((#‘𝑥) = 𝑗 ↔ (#‘𝑧) = 𝑗))
8887cbvrabv 3172 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗} = {𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}
8985, 88syl6eq 2660 . . . . . . 7 (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘} = {𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})
9089fveq2d 6107 . . . . . 6 (𝑘 = 𝑗 → (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9183, 90eqeq12d 2625 . . . . 5 (𝑘 = 𝑗 → (((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})))
9291cbvralv 3147 . . . 4 (∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
93 simpll 786 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
94 simplr 788 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ¬ 𝑧𝑦)
95 simprr 792 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9688fveq2i 6106 . . . . . . . . . 10 (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})
9796eqeq2i 2622 . . . . . . . . 9 (((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) ↔ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9897ralbii 2963 . . . . . . . 8 (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9995, 98sylibr 223 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}))
100 simprl 790 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
10193, 94, 99, 100hashbclem 13093 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘}))
102101expr 641 . . . . 5 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑘 ∈ ℤ) → (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}) → ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
103102ralrimdva 2952 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}) → ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
10492, 103syl5bi 231 . . 3 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) → ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
1058, 16, 24, 32, 82, 104findcard2s 8086 . 2 (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}))
106 oveq2 6557 . . . 4 (𝑘 = 𝐾 → ((#‘𝐴)C𝑘) = ((#‘𝐴)C𝐾))
107 eqeq2 2621 . . . . . 6 (𝑘 = 𝐾 → ((#‘𝑥) = 𝑘 ↔ (#‘𝑥) = 𝐾))
108107rabbidv 3164 . . . . 5 (𝑘 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})
109108fveq2d 6107 . . . 4 (𝑘 = 𝐾 → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
110106, 109eqeq12d 2625 . . 3 (𝑘 = 𝐾 → (((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})))
111110rspccva 3281 . 2 ((∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
112105, 111sylan 487 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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∪ cun 3538  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  Ccbc 12951  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-fac 12923  df-bc 12952  df-hash 12980 This theorem is referenced by:  hashbc2  15548  sylow1lem1  17836  musum  24717  ballotlem1  29875  ballotlem2  29877
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