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Theorem hashxplem 14333

Description: Lemma for hashxp 14334. (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 5647	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	fveq2d 6846	. . 3 ⊢ (𝑥 = ∅ → (♯‘(𝑥 × 𝐵)) = (♯‘(∅ × 𝐵)))
3		fveq2 6842	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq1d 7372	. . 3 ⊢ (𝑥 = ∅ → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘∅) · (♯‘𝐵)))
5	2, 4	eqeq12d 2752	. 2 ⊢ (𝑥 = ∅ → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))))
6		xpeq1 5647	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
7	6	fveq2d 6846	. . 3 ⊢ (𝑥 = 𝑦 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝑦 × 𝐵)))
8		fveq2 6842	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	oveq1d 7372	. . 3 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝑦) · (♯‘𝐵)))
10	7, 9	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝑦 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))))
11		xpeq1 5647	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∪ {𝑧}) × 𝐵))
12	11	fveq2d 6846	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝑥 × 𝐵)) = (♯‘((𝑦 ∪ {𝑧}) × 𝐵)))
13		fveq2 6842	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
14	13	oveq1d 7372	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
15	12, 14	eqeq12d 2752	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
16		xpeq1 5647	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
17	16	fveq2d 6846	. . 3 ⊢ (𝑥 = 𝐴 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝐴 × 𝐵)))
18		fveq2 6842	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
19	18	oveq1d 7372	. . 3 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
20	17, 19	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))))
21		hashxplem.1	. . . 4 ⊢ 𝐵 ∈ Fin
22		hashcl 14256	. . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
23	22	nn0cnd 12475	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
24	23	mul02d 11353	. . . 4 ⊢ (𝐵 ∈ Fin → (0 · (♯‘𝐵)) = 0)
25	21, 24	ax-mp 5	. . 3 ⊢ (0 · (♯‘𝐵)) = 0
26		hash0 14267	. . . 4 ⊢ (♯‘∅) = 0
27	26	oveq1i 7367	. . 3 ⊢ ((♯‘∅) · (♯‘𝐵)) = (0 · (♯‘𝐵))
28		0xp 5730	. . . . 5 ⊢ (∅ × 𝐵) = ∅
29	28	fveq2i 6845	. . . 4 ⊢ (♯‘(∅ × 𝐵)) = (♯‘∅)
30	29, 26	eqtri 2764	. . 3 ⊢ (♯‘(∅ × 𝐵)) = 0
31	25, 27, 30	3eqtr4ri 2775	. 2 ⊢ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))
32		oveq1 7364	. . . . 5 ⊢ ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
33	32	adantl 482	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
34		xpundir 5701	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) × 𝐵) = ((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))
35	34	fveq2i 6845	. . . . . 6 ⊢ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵)))
36		xpfi 9261	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)
37	21, 36	mpan2 689	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)
38		inxp 5788	. . . . . . . . 9 ⊢ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵))
39		disjsn 4672	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
40	39	biimpri 227	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
41	40	xpeq1d 5662	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = (∅ × (𝐵 ∩ 𝐵)))
42		0xp 5730	. . . . . . . . . 10 ⊢ (∅ × (𝐵 ∩ 𝐵)) = ∅
43	41, 42	eqtrdi 2792	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = ∅)
44	38, 43	eqtrid 2788	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅)
45		snfi 8988	. . . . . . . . . 10 ⊢ {𝑧} ∈ Fin
46		xpfi 9261	. . . . . . . . . 10 ⊢ (({𝑧} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ∈ Fin)
47	45, 21, 46	mp2an 690	. . . . . . . . 9 ⊢ ({𝑧} × 𝐵) ∈ Fin
48		hashun 14282	. . . . . . . . 9 ⊢ (((𝑦 × 𝐵) ∈ Fin ∧ ({𝑧} × 𝐵) ∈ Fin ∧ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
49	47, 48	mp3an2 1449	. . . . . . . 8 ⊢ (((𝑦 × 𝐵) ∈ Fin ∧ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
50	37, 44, 49	syl2an 596	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
51		snex 5388	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
52	21	elexi 3464	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
53	51, 52	xpcomen 9007	. . . . . . . . . 10 ⊢ ({𝑧} × 𝐵) ≈ (𝐵 × {𝑧})
54		vex 3449	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
55	52, 54	xpsnen 8999	. . . . . . . . . 10 ⊢ (𝐵 × {𝑧}) ≈ 𝐵
56	53, 55	entri 8948	. . . . . . . . 9 ⊢ ({𝑧} × 𝐵) ≈ 𝐵
57		hashen 14247	. . . . . . . . . 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 230	. . . . . . . 8 ⊢ (♯‘({𝑧} × 𝐵)) = (♯‘𝐵)
60	59	oveq2i 7368	. . . . . . 7 ⊢ ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵))
61	50, 60	eqtrdi 2792	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
62	35, 61	eqtrid 2788	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
63	62	adantr 481	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
64		hashunsng 14292	. . . . . . . 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 7372	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
67		hashcl 14256	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
68	67	nn0cnd 12475	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℂ)
69		ax-1cn 11109	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
70		nn0cn 12423	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
71	21, 22, 70	mp2b 10	. . . . . . . . . 10 ⊢ (♯‘𝐵) ∈ ℂ
72		adddir 11146	. . . . . . . . . 10 ⊢ (((♯‘𝑦) ∈ ℂ ∧ 1 ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))))
73	69, 71, 72	mp3an23 1453	. . . . . . . . 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 11160	. . . . . . . . 9 ⊢ (1 · (♯‘𝐵)) = (♯‘𝐵)
76	75	oveq2i 7368	. . . . . . . 8 ⊢ (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵))
77	74, 76	eqtrdi 2792	. . . . . . 7 ⊢ (𝑦 ∈ Fin → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
78	77	adantr 481	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
79	66, 78	eqtrd 2776	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
80	79	adantr 481	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
81	33, 63, 80	3eqtr4d 2786	. . 3 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
82	81	ex 413	. 2 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
83	5, 10, 15, 20, 31, 82	findcard2s 9109	1 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∪ cun 3908 ∩ cin 3909 ∅c0 4282 {csn 4586 class class class wbr 5105 × cxp 5631 ‘cfv 6496 (class class class)co 7357 ≈ cen 8880 Fincfn 8883 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ℕ₀cn0 12413 ♯chash 14230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231

This theorem is referenced by: hashxp 14334

W3C validator