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Theorem hashxp 10739

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 4618	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	fveq2d 5490	. . 3 ⊢ (𝑥 = ∅ → (♯‘(𝑥 × 𝐵)) = (♯‘(∅ × 𝐵)))
3		fveq2 5486	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq1d 5857	. . 3 ⊢ (𝑥 = ∅ → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘∅) · (♯‘𝐵)))
5	2, 4	eqeq12d 2180	. 2 ⊢ (𝑥 = ∅ → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))))
6		xpeq1 4618	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
7	6	fveq2d 5490	. . 3 ⊢ (𝑥 = 𝑦 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝑦 × 𝐵)))
8		fveq2 5486	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	oveq1d 5857	. . 3 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝑦) · (♯‘𝐵)))
10	7, 9	eqeq12d 2180	. 2 ⊢ (𝑥 = 𝑦 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))))
11		xpeq1 4618	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∪ {𝑧}) × 𝐵))
12	11	fveq2d 5490	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝑥 × 𝐵)) = (♯‘((𝑦 ∪ {𝑧}) × 𝐵)))
13		fveq2 5486	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
14	13	oveq1d 5857	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
15	12, 14	eqeq12d 2180	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
16		xpeq1 4618	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
17	16	fveq2d 5490	. . 3 ⊢ (𝑥 = 𝐴 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝐴 × 𝐵)))
18		fveq2 5486	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
19	18	oveq1d 5857	. . 3 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
20	17, 19	eqeq12d 2180	. 2 ⊢ (𝑥 = 𝐴 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))))
21		0xp 4684	. . . . 5 ⊢ (∅ × 𝐵) = ∅
22	21	fveq2i 5489	. . . 4 ⊢ (♯‘(∅ × 𝐵)) = (♯‘∅)
23		hash0 10710	. . . 4 ⊢ (♯‘∅) = 0
24	22, 23	eqtri 2186	. . 3 ⊢ (♯‘(∅ × 𝐵)) = 0
25	23	oveq1i 5852	. . . 4 ⊢ ((♯‘∅) · (♯‘𝐵)) = (0 · (♯‘𝐵))
26		hashcl 10694	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
27	26	nn0cnd 9169	. . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
28	27	mul02d 8290	. . . . 5 ⊢ (𝐵 ∈ Fin → (0 · (♯‘𝐵)) = 0)
29	28	adantl 275	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (0 · (♯‘𝐵)) = 0)
30	25, 29	syl5eq 2211	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘∅) · (♯‘𝐵)) = 0)
31	24, 30	eqtr4id 2218	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵)))
32		oveq1 5849	. . . . 5 ⊢ ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
33	32	adantl 275	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
34		xpundir 4661	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) × 𝐵) = ((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))
35	34	fveq2i 5489	. . . . . 6 ⊢ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵)))
36		simplr 520	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
37		simpllr 524	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐵 ∈ Fin)
38		xpfi 6895	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)
39	36, 37, 38	syl2anc 409	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 × 𝐵) ∈ Fin)
40		vex 2729	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
41		snfig 6780	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → {𝑧} ∈ Fin)
42	40, 41	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑧} ∈ Fin
43		xpfi 6895	. . . . . . . . . 10 ⊢ (({𝑧} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ∈ Fin)
44	42, 43	mpan 421	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → ({𝑧} × 𝐵) ∈ Fin)
45	44	ad3antlr 485	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ({𝑧} × 𝐵) ∈ Fin)
46		simprr 522	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
47	46	eldifbd 3128	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
48		inxp 4738	. . . . . . . . . 10 ⊢ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵))
49		disjsn 3638	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
50	49	biimpri 132	. . . . . . . . . . . 12 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
51	50	xpeq1d 4627	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = (∅ × (𝐵 ∩ 𝐵)))
52		0xp 4684	. . . . . . . . . . 11 ⊢ (∅ × (𝐵 ∩ 𝐵)) = ∅
53	51, 52	eqtrdi 2215	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = ∅)
54	48, 53	syl5eq 2211	. . . . . . . . 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 10718	. . . . . . . 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 1228	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
58	40	snex 4164	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ V
59	58	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → {𝑧} ∈ V)
60		xpcomeng 6794	. . . . . . . . . . 11 ⊢ (({𝑧} ∈ V ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ≈ (𝐵 × {𝑧}))
61	59, 37, 60	syl2anc 409	. . . . . . . . . 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 6788	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝑧 ∈ V) → (𝐵 × {𝑧}) ≈ 𝐵)
64	37, 62, 63	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐵 × {𝑧}) ≈ 𝐵)
65		entr 6750	. . . . . . . . . 10 ⊢ ((({𝑧} × 𝐵) ≈ (𝐵 × {𝑧}) ∧ (𝐵 × {𝑧}) ≈ 𝐵) → ({𝑧} × 𝐵) ≈ 𝐵)
66	61, 64, 65	syl2anc 409	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ({𝑧} × 𝐵) ≈ 𝐵)
67		hashen 10697	. . . . . . . . . 10 ⊢ ((({𝑧} × 𝐵) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵))
68	45, 37, 67	syl2anc 409	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵))
69	66, 68	mpbird 166	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘({𝑧} × 𝐵)) = (♯‘𝐵))
70	69	oveq2d 5858	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
71	57, 70	eqtrd 2198	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
72	35, 71	syl5eq 2211	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
73	72	adantr 274	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
74		hashunsng 10720	. . . . . . . . 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 5857	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
77	36, 47, 76	syl2anc 409	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
78		hashcl 10694	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
79	78	nn0cnd 9169	. . . . . . . 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 7925	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
83	77, 82	eqtrd 2198	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
84	83	adantr 274	. . . 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 2208	. . 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 114	. 2 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
87		simpl 108	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
88	5, 10, 15, 20, 31, 86, 87	findcard2sd 6858	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1343 ∈ wcel 2136 Vcvv 2726 ∖ cdif 3113 ∪ cun 3114 ∩ cin 3115 ⊆ wss 3116 ∅c0 3409 {csn 3576 class class class wbr 3982 × cxp 4602 ‘cfv 5188 (class class class)co 5842 ≈ cen 6704 Fincfn 6706 ℂcc 7751 0cc0 7753 1c1 7754 + caddc 7756 · cmul 7758 ♯chash 10688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-en 6707 df-dom 6708 df-fin 6709 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-fz 9945 df-ihash 10689

This theorem is referenced by: crth 12156 phimullem 12157

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