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Theorem hashxp 10897

Description: The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

**Assertion**
Ref	Expression
hashxp	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

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

Step	Hyp	Ref	Expression
1		xpeq1 4673	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	fveq2d 5558	. . 3 ⊢ (𝑥 = ∅ → (♯‘(𝑥 × 𝐵)) = (♯‘(∅ × 𝐵)))
3		fveq2 5554	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq1d 5933	. . 3 ⊢ (𝑥 = ∅ → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘∅) · (♯‘𝐵)))
5	2, 4	eqeq12d 2208	. 2 ⊢ (𝑥 = ∅ → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))))
6		xpeq1 4673	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
7	6	fveq2d 5558	. . 3 ⊢ (𝑥 = 𝑦 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝑦 × 𝐵)))
8		fveq2 5554	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	oveq1d 5933	. . 3 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝑦) · (♯‘𝐵)))
10	7, 9	eqeq12d 2208	. 2 ⊢ (𝑥 = 𝑦 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))))
11		xpeq1 4673	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∪ {𝑧}) × 𝐵))
12	11	fveq2d 5558	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝑥 × 𝐵)) = (♯‘((𝑦 ∪ {𝑧}) × 𝐵)))
13		fveq2 5554	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
14	13	oveq1d 5933	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
15	12, 14	eqeq12d 2208	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
16		xpeq1 4673	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
17	16	fveq2d 5558	. . 3 ⊢ (𝑥 = 𝐴 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝐴 × 𝐵)))
18		fveq2 5554	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
19	18	oveq1d 5933	. . 3 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
20	17, 19	eqeq12d 2208	. 2 ⊢ (𝑥 = 𝐴 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))))
21		0xp 4739	. . . . 5 ⊢ (∅ × 𝐵) = ∅
22	21	fveq2i 5557	. . . 4 ⊢ (♯‘(∅ × 𝐵)) = (♯‘∅)
23		hash0 10867	. . . 4 ⊢ (♯‘∅) = 0
24	22, 23	eqtri 2214	. . 3 ⊢ (♯‘(∅ × 𝐵)) = 0
25	23	oveq1i 5928	. . . 4 ⊢ ((♯‘∅) · (♯‘𝐵)) = (0 · (♯‘𝐵))
26		hashcl 10852	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
27	26	nn0cnd 9295	. . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
28	27	mul02d 8411	. . . . 5 ⊢ (𝐵 ∈ Fin → (0 · (♯‘𝐵)) = 0)
29	28	adantl 277	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (0 · (♯‘𝐵)) = 0)
30	25, 29	eqtrid 2238	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘∅) · (♯‘𝐵)) = 0)
31	24, 30	eqtr4id 2245	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵)))
32		oveq1 5925	. . . . 5 ⊢ ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
33	32	adantl 277	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
34		xpundir 4716	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) × 𝐵) = ((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))
35	34	fveq2i 5557	. . . . . 6 ⊢ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵)))
36		simplr 528	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
37		simpllr 534	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐵 ∈ Fin)
38		xpfi 6986	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)
39	36, 37, 38	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 × 𝐵) ∈ Fin)
40		vex 2763	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
41		snfig 6868	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → {𝑧} ∈ Fin)
42	40, 41	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑧} ∈ Fin
43		xpfi 6986	. . . . . . . . . 10 ⊢ (({𝑧} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ∈ Fin)
44	42, 43	mpan 424	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ({𝑧} × 𝐵) ∈ Fin)
45	44	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ({𝑧} × 𝐵) ∈ Fin)
46		simprr 531	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
47	46	eldifbd 3165	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
48		inxp 4796	. . . . . . . . . 10 ⊢ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵))
49		disjsn 3680	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
50	49	biimpri 133	. . . . . . . . . . . 12 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
51	50	xpeq1d 4682	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = (∅ × (𝐵 ∩ 𝐵)))
52		0xp 4739	. . . . . . . . . . 11 ⊢ (∅ × (𝐵 ∩ 𝐵)) = ∅
53	51, 52	eqtrdi 2242	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = ∅)
54	48, 53	eqtrid 2238	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅)
55	47, 54	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅)
56		hashun 10876	. . . . . . . 8 ⊢ (((𝑦 × 𝐵) ∈ Fin ∧ ({𝑧} × 𝐵) ∈ Fin ∧ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
57	39, 45, 55, 56	syl3anc 1249	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
58	40	snex 4214	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ V
59	58	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → {𝑧} ∈ V)
60		xpcomeng 6882	. . . . . . . . . . 11 ⊢ (({𝑧} ∈ V ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ≈ (𝐵 × {𝑧}))
61	59, 37, 60	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ({𝑧} × 𝐵) ≈ (𝐵 × {𝑧}))
62	40	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ V)
63		xpsneng 6876	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝑧 ∈ V) → (𝐵 × {𝑧}) ≈ 𝐵)
64	37, 62, 63	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐵 × {𝑧}) ≈ 𝐵)
65		entr 6838	. . . . . . . . . 10 ⊢ ((({𝑧} × 𝐵) ≈ (𝐵 × {𝑧}) ∧ (𝐵 × {𝑧}) ≈ 𝐵) → ({𝑧} × 𝐵) ≈ 𝐵)
66	61, 64, 65	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ({𝑧} × 𝐵) ≈ 𝐵)
67		hashen 10855	. . . . . . . . . 10 ⊢ ((({𝑧} × 𝐵) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵))
68	45, 37, 67	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵))
69	66, 68	mpbird 167	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘({𝑧} × 𝐵)) = (♯‘𝐵))
70	69	oveq2d 5934	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
71	57, 70	eqtrd 2226	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
72	35, 71	eqtrid 2238	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
73	72	adantr 276	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
74		hashunsng 10878	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
75	40, 74	ax-mp 5	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
76	75	oveq1d 5933	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
77	36, 47, 76	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
78		hashcl 10852	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
79	78	nn0cnd 9295	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℂ)
80	36, 79	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝑦) ∈ ℂ)
81	37, 27	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘𝐵) ∈ ℂ)
82	80, 81	adddirp1d 8046	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
83	77, 82	eqtrd 2226	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
84	83	adantr 276	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
85	33, 73, 84	3eqtr4d 2236	. . 3 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
86	85	ex 115	. 2 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
87		simpl 109	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
88	5, 10, 15, 20, 31, 86, 87	findcard2sd 6948	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 Vcvv 2760 ∖ cdif 3150 ∪ cun 3151 ∩ cin 3152 ⊆ wss 3153 ∅c0 3446 {csn 3618 class class class wbr 4029 × cxp 4657 ‘cfv 5254 (class class class)co 5918 ≈ cen 6792 Fincfn 6794 ℂcc 7870 0cc0 7872 1c1 7873 + caddc 7875 · cmul 7877 ♯chash 10846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-fz 10075 df-ihash 10847

This theorem is referenced by: crth 12362 phimullem 12363

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