hashmap - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hashmap		Structured version Visualization version GIF version

Theorem hashmap 14462

Description: The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.) (Proof shortened by AV, 18-Jul-2022.)

**Assertion**
Ref	Expression
hashmap	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Proof of Theorem hashmap Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7408	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m ∅))
2	1	fveq2d 6875	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m ∅)))
3		fveq2 6871	. . . . . 6 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq2d 7416	. . . . 5 ⊢ (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
5	2, 4	eqeq12d 2781	. . . 4 ⊢ (𝑥 = ∅ → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅))))
6	5	imbi2d 343	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))))
7		oveq2 7408	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝑦))
8	7	fveq2d 6875	. . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝑦)))
9		fveq2 6871	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
10	9	oveq2d 7416	. . . . 5 ⊢ (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
11	8, 10	eqeq12d 2781	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))))
12	11	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)))))
13		oveq2 7408	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m (𝑦 ∪ {𝑧})))
14	13	fveq2d 6875	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))))
15		fveq2 6871	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq2d 7416	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
17	14, 16	eqeq12d 2781	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
18	17	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
19		oveq2 7408	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝐵))
20	19	fveq2d 6875	. . . . 5 ⊢ (𝑥 = 𝐵 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝐵)))
21		fveq2 6871	. . . . . 6 ⊢ (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
22	21	oveq2d 7416	. . . . 5 ⊢ (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
23	20, 22	eqeq12d 2781	. . . 4 ⊢ (𝑥 = 𝐵 → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
24	23	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))))
25		hashcl 14383	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
26	25	nn0cnd 12558	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
27	26	exp0d 14167	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28		hash0 14394	. . . . . 6 ⊢ (♯‘∅) = 0
29	28	oveq2i 7411	. . . . 5 ⊢ ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
30	29	a1i 11	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31		mapdm0 8827	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ↑_m ∅) = {∅})
32	31	fveq2d 6875	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = (♯‘{∅}))
33		0ex 5262	. . . . . 6 ⊢ ∅ ∈ V
34		hashsng 14396	. . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
35	33, 34	mp1i 14	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{∅}) = 1)
36	32, 35	eqtrd 2800	. . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = 1)
37	27, 30, 36	3eqtr4rd 2811	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38		oveq1 7407	. . . . . 6 ⊢ ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39		vex 3461	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
40	39	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ V)
41		vsnex 5397	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
42	41	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → {𝑧} ∈ V)
43		elex 3478	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
44	43	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ V)
45		simprr 784	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
46		disjsn 4673	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
47	45, 46	sylibr 237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48		mapunen 9122	. . . . . . . . . 10 ⊢ (((𝑦 ∈ V ∧ {𝑧} ∈ V ∧ 𝐴 ∈ V) ∧ (𝑦 ∩ {𝑧}) = ∅) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})))
49	40, 42, 44, 47, 48	syl31anc 1396	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})))
50		simpl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ Fin)
51		simprl 782	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
52		snfi 9028	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
53		unfi 9143	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
54	51, 52, 53	sylancl 597	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
55		mapfi 9293	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
56	50, 54, 55	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
57		mapfi 9293	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴 ↑_m 𝑦) ∈ Fin)
58	57	adantrr 729	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m 𝑦) ∈ Fin)
59		mapfi 9293	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ↑_m {𝑧}) ∈ Fin)
60	50, 52, 59	sylancl 597	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ∈ Fin)
61		xpfi 9267	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m 𝑦) ∈ Fin ∧ (𝐴 ↑_m {𝑧}) ∈ Fin) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
62	58, 60, 61	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
63		hashen 14374	. . . . . . . . . 10 ⊢ (((𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin ∧ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) ↔ (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
64	56, 62, 63	syl2anc 595	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) ↔ (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
65	49, 64	mpbird 260	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
66		hashxp 14461	. . . . . . . . 9 ⊢ (((𝐴 ↑_m 𝑦) ∈ Fin ∧ (𝐴 ↑_m {𝑧}) ∈ Fin) → (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))))
67	58, 60, 66	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))))
68		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
69	68	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
70	50, 69	mapsnend 9021	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ≈ 𝐴)
71		hashen 14374	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
72	60, 50, 71	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
73	70, 72	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴))
74	73	oveq2d 7416	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
75	65, 67, 74	3eqtrd 2804	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
76		hashunsng 14419	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
77	76	elv 3462	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
78	77	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
79	78	oveq2d 7416	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑((♯‘𝑦) + 1)))
80	26	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝐴) ∈ ℂ)
81		hashcl 14383	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
82	81	ad2antrl 740	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝑦) ∈ ℕ₀)
83	80, 82	expp1d 14174	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
84	79, 83	eqtrd 2800	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
85	75, 84	eqeq12d 2781	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) ↔ ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴))))
86	38, 85	imbitrrid 249	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
87	86	expcom 418	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐴 ∈ Fin → ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
88	87	a2d 30	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))) → (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
89	6, 12, 18, 24, 37, 88	findcard2s 9138	. 2 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
90	89	impcom 412	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ∪ cun 3905 ∩ cin 3906 ∅c0 4288 {csn 4585 class class class wbr 5105 × cxp 5650 ‘cfv 6525 (class class class)co 7400 ↑_m cmap 8812 ≈ cen 8928 Fincfn 8931 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 ℕ₀cn0 12495 ↑cexp 14088 ♯chash 14357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-oadd 8445 df-er 8682 df-map 8814 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-dju 9875 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-seq 14029 df-exp 14089 df-hash 14358

This theorem is referenced by: hashpw 14463 hashwrdn 14574 prmreclem2 16967 efmndhash 18925 birthdaylem2 27075

W3C validator