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Theorem hashmap 13799

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 7167	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m ∅))
2	1	fveq2d 6677	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m ∅)))
3		fveq2 6673	. . . . . 6 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq2d 7175	. . . . 5 ⊢ (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
5	2, 4	eqeq12d 2840	. . . 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 7167	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝑦))
8	7	fveq2d 6677	. . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝑦)))
9		fveq2 6673	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
10	9	oveq2d 7175	. . . . 5 ⊢ (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
11	8, 10	eqeq12d 2840	. . . 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 7167	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m (𝑦 ∪ {𝑧})))
14	13	fveq2d 6677	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))))
15		fveq2 6673	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq2d 7175	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
17	14, 16	eqeq12d 2840	. . . 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 7167	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝐵))
20	19	fveq2d 6677	. . . . 5 ⊢ (𝑥 = 𝐵 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝐵)))
21		fveq2 6673	. . . . . 6 ⊢ (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
22	21	oveq2d 7175	. . . . 5 ⊢ (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
23	20, 22	eqeq12d 2840	. . . 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 13720	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
26	25	nn0cnd 11960	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
27	26	exp0d 13507	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28		hash0 13731	. . . . . 6 ⊢ (♯‘∅) = 0
29	28	oveq2i 7170	. . . . 5 ⊢ ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
30	29	a1i 11	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31		mapdm0 8424	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ↑_m ∅) = {∅})
32	31	fveq2d 6677	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = (♯‘{∅}))
33		0ex 5214	. . . . . 6 ⊢ ∅ ∈ V
34		hashsng 13733	. . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
35	33, 34	mp1i 13	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{∅}) = 1)
36	32, 35	eqtrd 2859	. . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = 1)
37	27, 30, 36	3eqtr4rd 2870	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38		oveq1 7166	. . . . . 6 ⊢ ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39		vex 3500	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
40	39	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ V)
41		snex 5335	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
42	41	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → {𝑧} ∈ V)
43		elex 3515	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
44	43	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ V)
45		simprr 771	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
46		disjsn 4650	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
47	45, 46	sylibr 236	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48		mapunen 8689	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})))
50		simpl 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ Fin)
51		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
52		snfi 8597	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
53		unfi 8788	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
54	51, 52, 53	sylancl 588	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
55		mapfi 8823	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
56	50, 54, 55	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
57		mapfi 8823	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴 ↑_m 𝑦) ∈ Fin)
58	57	adantrr 715	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m 𝑦) ∈ Fin)
59		mapfi 8823	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ↑_m {𝑧}) ∈ Fin)
60	50, 52, 59	sylancl 588	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ∈ Fin)
61		xpfi 8792	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m 𝑦) ∈ Fin ∧ (𝐴 ↑_m {𝑧}) ∈ Fin) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
62	58, 60, 61	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
63		hashen 13710	. . . . . . . . . 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 586	. . . . . . . . 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 259	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
66		hashxp 13798	. . . . . . . . 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 586	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))))
68		vex 3500	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
69	68	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
70	50, 69	mapsnend 8591	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ≈ 𝐴)
71		hashen 13710	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
72	60, 50, 71	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
73	70, 72	mpbird 259	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴))
74	73	oveq2d 7175	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
75	65, 67, 74	3eqtrd 2863	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
76		hashunsng 13756	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
77	76	elv 3502	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
78	77	adantl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
79	78	oveq2d 7175	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑((♯‘𝑦) + 1)))
80	26	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝐴) ∈ ℂ)
81		hashcl 13720	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
82	81	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝑦) ∈ ℕ₀)
83	80, 82	expp1d 13514	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
84	79, 83	eqtrd 2859	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
85	75, 84	eqeq12d 2840	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) ↔ ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴))))
86	38, 85	syl5ibr 248	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
87	86	expcom 416	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐴 ∈ Fin → ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
88	87	a2d 29	. . 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 8762	. 2 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
90	89	impcom 410	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∪ cun 3937 ∩ cin 3938 ∅c0 4294 {csn 4570 class class class wbr 5069 × cxp 5556 ‘cfv 6358 (class class class)co 7159 ↑_m cmap 8409 ≈ cen 8509 Fincfn 8512 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 ℕ₀cn0 11900 ↑cexp 13432 ♯chash 13693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 df-exp 13433 df-hash 13694

This theorem is referenced by: hashpw 13800 hashwrdn 13901 prmreclem2 16256 efmndhash 18044 birthdaylem2 25533

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