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Theorem hashxplem 13797

Description: Lemma for hashxp 13798. (Contributed by Paul Chapman, 30-Nov-2012.)

**Hypothesis**
Ref	Expression
hashxplem.1	⊢ 𝐵 ∈ Fin

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

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

Step	Hyp	Ref	Expression
1		xpeq1 5572	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	fveq2d 6677	. . 3 ⊢ (𝑥 = ∅ → (♯‘(𝑥 × 𝐵)) = (♯‘(∅ × 𝐵)))
3		fveq2 6673	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq1d 7174	. . 3 ⊢ (𝑥 = ∅ → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘∅) · (♯‘𝐵)))
5	2, 4	eqeq12d 2840	. 2 ⊢ (𝑥 = ∅ → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))))
6		xpeq1 5572	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
7	6	fveq2d 6677	. . 3 ⊢ (𝑥 = 𝑦 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝑦 × 𝐵)))
8		fveq2 6673	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	oveq1d 7174	. . 3 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝑦) · (♯‘𝐵)))
10	7, 9	eqeq12d 2840	. 2 ⊢ (𝑥 = 𝑦 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))))
11		xpeq1 5572	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∪ {𝑧}) × 𝐵))
12	11	fveq2d 6677	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝑥 × 𝐵)) = (♯‘((𝑦 ∪ {𝑧}) × 𝐵)))
13		fveq2 6673	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
14	13	oveq1d 7174	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
15	12, 14	eqeq12d 2840	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
16		xpeq1 5572	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
17	16	fveq2d 6677	. . 3 ⊢ (𝑥 = 𝐴 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝐴 × 𝐵)))
18		fveq2 6673	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
19	18	oveq1d 7174	. . 3 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
20	17, 19	eqeq12d 2840	. 2 ⊢ (𝑥 = 𝐴 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))))
21		hashxplem.1	. . . 4 ⊢ 𝐵 ∈ Fin
22		hashcl 13720	. . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
23	22	nn0cnd 11960	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
24	23	mul02d 10841	. . . 4 ⊢ (𝐵 ∈ Fin → (0 · (♯‘𝐵)) = 0)
25	21, 24	ax-mp 5	. . 3 ⊢ (0 · (♯‘𝐵)) = 0
26		hash0 13731	. . . 4 ⊢ (♯‘∅) = 0
27	26	oveq1i 7169	. . 3 ⊢ ((♯‘∅) · (♯‘𝐵)) = (0 · (♯‘𝐵))
28		0xp 5652	. . . . 5 ⊢ (∅ × 𝐵) = ∅
29	28	fveq2i 6676	. . . 4 ⊢ (♯‘(∅ × 𝐵)) = (♯‘∅)
30	29, 26	eqtri 2847	. . 3 ⊢ (♯‘(∅ × 𝐵)) = 0
31	25, 27, 30	3eqtr4ri 2858	. 2 ⊢ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))
32		oveq1 7166	. . . . 5 ⊢ ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
33	32	adantl 484	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
34		xpundir 5624	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) × 𝐵) = ((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))
35	34	fveq2i 6676	. . . . . 6 ⊢ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵)))
36		xpfi 8792	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)
37	21, 36	mpan2 689	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)
38		inxp 5706	. . . . . . . . 9 ⊢ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵))
39		disjsn 4650	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
40	39	biimpri 230	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
41	40	xpeq1d 5587	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = (∅ × (𝐵 ∩ 𝐵)))
42		0xp 5652	. . . . . . . . . 10 ⊢ (∅ × (𝐵 ∩ 𝐵)) = ∅
43	41, 42	syl6eq 2875	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = ∅)
44	38, 43	syl5eq 2871	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅)
45		snfi 8597	. . . . . . . . . 10 ⊢ {𝑧} ∈ Fin
46		xpfi 8792	. . . . . . . . . 10 ⊢ (({𝑧} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ∈ Fin)
47	45, 21, 46	mp2an 690	. . . . . . . . 9 ⊢ ({𝑧} × 𝐵) ∈ Fin
48		hashun 13746	. . . . . . . . 9 ⊢ (((𝑦 × 𝐵) ∈ Fin ∧ ({𝑧} × 𝐵) ∈ Fin ∧ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
49	47, 48	mp3an2 1445	. . . . . . . 8 ⊢ (((𝑦 × 𝐵) ∈ Fin ∧ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
50	37, 44, 49	syl2an 597	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
51		snex 5335	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
52	21	elexi 3516	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
53	51, 52	xpcomen 8611	. . . . . . . . . 10 ⊢ ({𝑧} × 𝐵) ≈ (𝐵 × {𝑧})
54		vex 3500	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
55	52, 54	xpsnen 8604	. . . . . . . . . 10 ⊢ (𝐵 × {𝑧}) ≈ 𝐵
56	53, 55	entri 8566	. . . . . . . . 9 ⊢ ({𝑧} × 𝐵) ≈ 𝐵
57		hashen 13710	. . . . . . . . . 10 ⊢ ((({𝑧} × 𝐵) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵))
58	47, 21, 57	mp2an 690	. . . . . . . . 9 ⊢ ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵)
59	56, 58	mpbir 233	. . . . . . . 8 ⊢ (♯‘({𝑧} × 𝐵)) = (♯‘𝐵)
60	59	oveq2i 7170	. . . . . . 7 ⊢ ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵))
61	50, 60	syl6eq 2875	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
62	35, 61	syl5eq 2871	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
63	62	adantr 483	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
64		hashunsng 13756	. . . . . . . 8 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
65	54, 64	ax-mp 5	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
66	65	oveq1d 7174	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
67		hashcl 13720	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
68	67	nn0cnd 11960	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℂ)
69		ax-1cn 10598	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
70		nn0cn 11910	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
71	21, 22, 70	mp2b 10	. . . . . . . . . 10 ⊢ (♯‘𝐵) ∈ ℂ
72		adddir 10635	. . . . . . . . . 10 ⊢ (((♯‘𝑦) ∈ ℂ ∧ 1 ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))))
73	69, 71, 72	mp3an23 1449	. . . . . . . . 9 ⊢ ((♯‘𝑦) ∈ ℂ → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))))
74	68, 73	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))))
75	71	mulid2i 10649	. . . . . . . . 9 ⊢ (1 · (♯‘𝐵)) = (♯‘𝐵)
76	75	oveq2i 7170	. . . . . . . 8 ⊢ (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵))
77	74, 76	syl6eq 2875	. . . . . . 7 ⊢ (𝑦 ∈ Fin → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
78	77	adantr 483	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
79	66, 78	eqtrd 2859	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
80	79	adantr 483	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
81	33, 63, 80	3eqtr4d 2869	. . 3 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
82	81	ex 415	. 2 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
83	5, 10, 15, 20, 31, 82	findcard2s 8762	1 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 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 ≈ cen 8509 Fincfn 8512 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 ℕ₀cn0 11900 ♯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-oadd 8109 df-er 8292 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-hash 13694

This theorem is referenced by: hashxp 13798

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