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Theorem hashmap 14438

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 7393	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m ∅))
2	1	fveq2d 6860	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m ∅)))
3		fveq2 6856	. . . . . 6 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq2d 7401	. . . . 5 ⊢ (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
5	2, 4	eqeq12d 2772	. . . 4 ⊢ (𝑥 = ∅ → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅))))
6	5	imbi2d 342	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))))
7		oveq2 7393	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝑦))
8	7	fveq2d 6860	. . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝑦)))
9		fveq2 6856	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
10	9	oveq2d 7401	. . . . 5 ⊢ (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
11	8, 10	eqeq12d 2772	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))))
12	11	imbi2d 342	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)))))
13		oveq2 7393	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m (𝑦 ∪ {𝑧})))
14	13	fveq2d 6860	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))))
15		fveq2 6856	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq2d 7401	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
17	14, 16	eqeq12d 2772	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
18	17	imbi2d 342	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
19		oveq2 7393	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝐵))
20	19	fveq2d 6860	. . . . 5 ⊢ (𝑥 = 𝐵 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝐵)))
21		fveq2 6856	. . . . . 6 ⊢ (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
22	21	oveq2d 7401	. . . . 5 ⊢ (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
23	20, 22	eqeq12d 2772	. . . 4 ⊢ (𝑥 = 𝐵 → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
24	23	imbi2d 342	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))))
25		hashcl 14359	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
26	25	nn0cnd 12534	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
27	26	exp0d 14143	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28		hash0 14370	. . . . . 6 ⊢ (♯‘∅) = 0
29	28	oveq2i 7396	. . . . 5 ⊢ ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
30	29	a1i 11	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31		mapdm0 8812	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ↑_m ∅) = {∅})
32	31	fveq2d 6860	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = (♯‘{∅}))
33		0ex 5251	. . . . . 6 ⊢ ∅ ∈ V
34		hashsng 14372	. . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
35	33, 34	mp1i 13	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{∅}) = 1)
36	32, 35	eqtrd 2791	. . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = 1)
37	27, 30, 36	3eqtr4rd 2802	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38		oveq1 7392	. . . . . 6 ⊢ ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39		vex 3452	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
40	39	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ V)
41		vsnex 5386	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
42	41	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → {𝑧} ∈ V)
43		elex 3469	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
44	43	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ V)
45		simprr 780	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
46		disjsn 4664	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
47	45, 46	sylibr 236	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48		mapunen 9107	. . . . . . . . . 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 1388	. . . . . . . . 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 778	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
52		snfi 9013	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
53		unfi 9128	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
54	51, 52, 53	sylancl 594	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
55		mapfi 9281	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
56	50, 54, 55	syl2anc 592	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
57		mapfi 9281	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴 ↑_m 𝑦) ∈ Fin)
58	57	adantrr 725	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m 𝑦) ∈ Fin)
59		mapfi 9281	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ↑_m {𝑧}) ∈ Fin)
60	50, 52, 59	sylancl 594	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ∈ Fin)
61		xpfi 9253	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m 𝑦) ∈ Fin ∧ (𝐴 ↑_m {𝑧}) ∈ Fin) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
62	58, 60, 61	syl2anc 592	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
63		hashen 14350	. . . . . . . . . 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 592	. . . . . . . . 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 14437	. . . . . . . . 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 592	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))))
68		vex 3452	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
69	68	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
70	50, 69	mapsnend 9006	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ≈ 𝐴)
71		hashen 14350	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
72	60, 50, 71	syl2anc 592	. . . . . . . . . 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 7401	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
75	65, 67, 74	3eqtrd 2795	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
76		hashunsng 14395	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
77	76	elv 3453	. . . . . . . . . 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 7401	. . . . . . . 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 14359	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
82	81	ad2antrl 736	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝑦) ∈ ℕ₀)
83	80, 82	expp1d 14150	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
84	79, 83	eqtrd 2791	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
85	75, 84	eqeq12d 2772	. . . . . 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 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 9123	. 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 1554 ∈ wcel 2136 Vcvv 3448 ∪ cun 3897 ∩ cin 3898 ∅c0 4280 {csn 4576 class class class wbr 5094 × cxp 5638 ‘cfv 6510 (class class class)co 7385 ↑_m cmap 8796 ≈ cen 8913 Fincfn 8916 ℂcc 11061 0cc0 11063 1c1 11064 + caddc 11066 · cmul 11068 ℕ₀cn0 12471 ↑cexp 14064 ♯chash 14333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-oadd 8429 df-er 8666 df-map 8798 df-pm 8799 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-dju 9849 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-nn 12201 df-n0 12472 df-z 12559 df-uz 12830 df-fz 13503 df-seq 14005 df-exp 14065 df-hash 14334

This theorem is referenced by: hashpw 14439 hashwrdn 14550 prmreclem2 16929 efmndhash 18886 birthdaylem2 26987

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