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Theorem hashxplem 14472

Description: Lemma for hashxp 14473. (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 5699	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	fveq2d 6910	. . 3 ⊢ (𝑥 = ∅ → (♯‘(𝑥 × 𝐵)) = (♯‘(∅ × 𝐵)))
3		fveq2 6906	. . . 4 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq1d 7446	. . 3 ⊢ (𝑥 = ∅ → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘∅) · (♯‘𝐵)))
5	2, 4	eqeq12d 2753	. 2 ⊢ (𝑥 = ∅ → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))))
6		xpeq1 5699	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
7	6	fveq2d 6910	. . 3 ⊢ (𝑥 = 𝑦 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝑦 × 𝐵)))
8		fveq2 6906	. . . 4 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	oveq1d 7446	. . 3 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝑦) · (♯‘𝐵)))
10	7, 9	eqeq12d 2753	. 2 ⊢ (𝑥 = 𝑦 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))))
11		xpeq1 5699	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∪ {𝑧}) × 𝐵))
12	11	fveq2d 6910	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝑥 × 𝐵)) = (♯‘((𝑦 ∪ {𝑧}) × 𝐵)))
13		fveq2 6906	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
14	13	oveq1d 7446	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
15	12, 14	eqeq12d 2753	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
16		xpeq1 5699	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
17	16	fveq2d 6910	. . 3 ⊢ (𝑥 = 𝐴 → (♯‘(𝑥 × 𝐵)) = (♯‘(𝐴 × 𝐵)))
18		fveq2 6906	. . . 4 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
19	18	oveq1d 7446	. . 3 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) · (♯‘𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
20	17, 19	eqeq12d 2753	. 2 ⊢ (𝑥 = 𝐴 → ((♯‘(𝑥 × 𝐵)) = ((♯‘𝑥) · (♯‘𝐵)) ↔ (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵))))
21		hashxplem.1	. . . 4 ⊢ 𝐵 ∈ Fin
22		hashcl 14395	. . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
23	22	nn0cnd 12589	. . . . 5 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
24	23	mul02d 11459	. . . 4 ⊢ (𝐵 ∈ Fin → (0 · (♯‘𝐵)) = 0)
25	21, 24	ax-mp 5	. . 3 ⊢ (0 · (♯‘𝐵)) = 0
26		hash0 14406	. . . 4 ⊢ (♯‘∅) = 0
27	26	oveq1i 7441	. . 3 ⊢ ((♯‘∅) · (♯‘𝐵)) = (0 · (♯‘𝐵))
28		0xp 5784	. . . . 5 ⊢ (∅ × 𝐵) = ∅
29	28	fveq2i 6909	. . . 4 ⊢ (♯‘(∅ × 𝐵)) = (♯‘∅)
30	29, 26	eqtri 2765	. . 3 ⊢ (♯‘(∅ × 𝐵)) = 0
31	25, 27, 30	3eqtr4ri 2776	. 2 ⊢ (♯‘(∅ × 𝐵)) = ((♯‘∅) · (♯‘𝐵))
32		oveq1 7438	. . . . 5 ⊢ ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
33	32	adantl 481	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
34		xpundir 5755	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) × 𝐵) = ((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))
35	34	fveq2i 6909	. . . . . 6 ⊢ (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵)))
36		xpfi 9358	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)
37	21, 36	mpan2 691	. . . . . . . 8 ⊢ (𝑦 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)
38		inxp 5842	. . . . . . . . 9 ⊢ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵))
39		disjsn 4711	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
40	39	biimpri 228	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
41	40	xpeq1d 5714	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = (∅ × (𝐵 ∩ 𝐵)))
42		0xp 5784	. . . . . . . . . 10 ⊢ (∅ × (𝐵 ∩ 𝐵)) = ∅
43	41, 42	eqtrdi 2793	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) × (𝐵 ∩ 𝐵)) = ∅)
44	38, 43	eqtrid 2789	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅)
45		snfi 9083	. . . . . . . . . 10 ⊢ {𝑧} ∈ Fin
46		xpfi 9358	. . . . . . . . . 10 ⊢ (({𝑧} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑧} × 𝐵) ∈ Fin)
47	45, 21, 46	mp2an 692	. . . . . . . . 9 ⊢ ({𝑧} × 𝐵) ∈ Fin
48		hashun 14421	. . . . . . . . 9 ⊢ (((𝑦 × 𝐵) ∈ Fin ∧ ({𝑧} × 𝐵) ∈ Fin ∧ ((𝑦 × 𝐵) ∩ ({𝑧} × 𝐵)) = ∅) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))))
49	47, 48	mp3an2 1451	. . . . . . . 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 5436	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
52	21	elexi 3503	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
53	51, 52	xpcomen 9103	. . . . . . . . . 10 ⊢ ({𝑧} × 𝐵) ≈ (𝐵 × {𝑧})
54		vex 3484	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
55	52, 54	xpsnen 9095	. . . . . . . . . 10 ⊢ (𝐵 × {𝑧}) ≈ 𝐵
56	53, 55	entri 9048	. . . . . . . . 9 ⊢ ({𝑧} × 𝐵) ≈ 𝐵
57		hashen 14386	. . . . . . . . . 10 ⊢ ((({𝑧} × 𝐵) ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵))
58	47, 21, 57	mp2an 692	. . . . . . . . 9 ⊢ ((♯‘({𝑧} × 𝐵)) = (♯‘𝐵) ↔ ({𝑧} × 𝐵) ≈ 𝐵)
59	56, 58	mpbir 231	. . . . . . . 8 ⊢ (♯‘({𝑧} × 𝐵)) = (♯‘𝐵)
60	59	oveq2i 7442	. . . . . . 7 ⊢ ((♯‘(𝑦 × 𝐵)) + (♯‘({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵))
61	50, 60	eqtrdi 2793	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 × 𝐵) ∪ ({𝑧} × 𝐵))) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
62	35, 61	eqtrid 2789	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
63	62	adantr 480	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 × 𝐵)) + (♯‘𝐵)))
64		hashunsng 14431	. . . . . . . 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 7446	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) + 1) · (♯‘𝐵)))
67		hashcl 14395	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
68	67	nn0cnd 12589	. . . . . . . . 9 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℂ)
69		ax-1cn 11213	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
70		nn0cn 12536	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
71	21, 22, 70	mp2b 10	. . . . . . . . . 10 ⊢ (♯‘𝐵) ∈ ℂ
72		adddir 11252	. . . . . . . . . 10 ⊢ (((♯‘𝑦) ∈ ℂ ∧ 1 ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))))
73	69, 71, 72	mp3an23 1455	. . . . . . . . 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	mullidi 11266	. . . . . . . . 9 ⊢ (1 · (♯‘𝐵)) = (♯‘𝐵)
76	75	oveq2i 7442	. . . . . . . 8 ⊢ (((♯‘𝑦) · (♯‘𝐵)) + (1 · (♯‘𝐵))) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵))
77	74, 76	eqtrdi 2793	. . . . . . 7 ⊢ (𝑦 ∈ Fin → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
78	77	adantr 480	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((♯‘𝑦) + 1) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
79	66, 78	eqtrd 2777	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
80	79	adantr 480	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)) = (((♯‘𝑦) · (♯‘𝐵)) + (♯‘𝐵)))
81	33, 63, 80	3eqtr4d 2787	. . 3 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵))) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵)))
82	81	ex 412	. 2 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((♯‘(𝑦 × 𝐵)) = ((♯‘𝑦) · (♯‘𝐵)) → (♯‘((𝑦 ∪ {𝑧}) × 𝐵)) = ((♯‘(𝑦 ∪ {𝑧})) · (♯‘𝐵))))
83	5, 10, 15, 20, 31, 82	findcard2s 9205	1 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ≈ cen 8982 Fincfn 8985 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕ₀cn0 12526 ♯chash 14369

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370

This theorem is referenced by: hashxp 14473

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