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Theorem divalglem8 16448

Description: Lemma for divalg 16451. (Contributed by Paul Chapman, 21-Mar-2011.)

**Hypotheses**
Ref	Expression
divalglem8.1	⊢ 𝑁 ∈ ℤ
divalglem8.2	⊢ 𝐷 ∈ ℤ
divalglem8.3	⊢ 𝐷 ≠ 0
divalglem8.4	⊢ 𝑆 = {𝑟 ∈ ℕ₀ ∣ 𝐷 ∥ (𝑁 − 𝑟)}

**Assertion**
Ref	Expression
divalglem8	⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑋 < (abs‘𝐷) ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌)))

Distinct variable groups: 𝐷,𝑟 𝑁,𝑟 Allowed substitution hints: 𝑆(𝑟) 𝐾(𝑟) 𝑋(𝑟) 𝑌(𝑟)

Proof of Theorem divalglem8 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divalglem8.4	. . . . . . . . . . . . 13 ⊢ 𝑆 = {𝑟 ∈ ℕ₀ ∣ 𝐷 ∥ (𝑁 − 𝑟)}
2	1	ssrab3 4105	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ ℕ₀
3		nn0sscn 12558	. . . . . . . . . . . 12 ⊢ ℕ₀ ⊆ ℂ
4	2, 3	sstri 4018	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ℂ
5	4	sseli 4004	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑆 → 𝑌 ∈ ℂ)
6	4	sseli 4004	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ ℂ)
7		divalglem8.2	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ ℤ
8		divalglem8.3	. . . . . . . . . . . . . 14 ⊢ 𝐷 ≠ 0
9		nnabscl 15374	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ)
10	7, 8, 9	mp2an 691	. . . . . . . . . . . . 13 ⊢ (abs‘𝐷) ∈ ℕ
11	10	nnzi 12667	. . . . . . . . . . . 12 ⊢ (abs‘𝐷) ∈ ℤ
12		zmulcl 12692	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐾 · (abs‘𝐷)) ∈ ℤ)
13	11, 12	mpan2 690	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℤ)
14	13	zcnd 12748	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℂ)
15		subadd 11539	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (𝐾 · (abs‘𝐷)) ∈ ℂ) → ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) = 𝑌))
16	5, 6, 14, 15	syl3an 1160	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) = 𝑌))
17	16	3com12 1123	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) = 𝑌))
18		eqcom 2747	. . . . . . . 8 ⊢ ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋))
19		eqcom 2747	. . . . . . . 8 ⊢ ((𝑋 + (𝐾 · (abs‘𝐷))) = 𝑌 ↔ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))))
20	17, 18, 19	3bitr3g 313	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))))
21	20	3adant1r 1177	. . . . . 6 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ 𝑌 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))))
22	21	3adant2r 1179	. . . . 5 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))))
23		breq1 5169	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑌 → (𝑧 < (abs‘𝐷) ↔ 𝑌 < (abs‘𝐷)))
24		eleq1 2832	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑌 → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ 𝑌 ∈ (0...((abs‘𝐷) − 1))))
25	23, 24	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑌 → ((𝑧 < (abs‘𝐷) → 𝑧 ∈ (0...((abs‘𝐷) − 1))) ↔ (𝑌 < (abs‘𝐷) → 𝑌 ∈ (0...((abs‘𝐷) − 1)))))
26	2	sseli 4004	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℕ₀)
27		elnn0z 12652	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ₀ ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧))
28	26, 27	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧))
29	28	anim1i 614	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → ((𝑧 ∈ ℤ ∧ 0 ≤ 𝑧) ∧ 𝑧 < (abs‘𝐷)))
30		df-3an 1089	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)) ↔ ((𝑧 ∈ ℤ ∧ 0 ≤ 𝑧) ∧ 𝑧 < (abs‘𝐷)))
31	29, 30	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)))
32		0z 12650	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
33		elfzm11 13655	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷))))
34	32, 11, 33	mp2an 691	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)))
35	31, 34	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → 𝑧 ∈ (0...((abs‘𝐷) − 1)))
36	35	ex 412	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑆 → (𝑧 < (abs‘𝐷) → 𝑧 ∈ (0...((abs‘𝐷) − 1))))
37	25, 36	vtoclga 3589	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑆 → (𝑌 < (abs‘𝐷) → 𝑌 ∈ (0...((abs‘𝐷) − 1))))
38		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑌 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
39	38	biimpd 229	. . . . . . . . . 10 ⊢ (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑌 ∈ (0...((abs‘𝐷) − 1)) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
40	37, 39	sylan9 507	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝑆 ∧ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))) → (𝑌 < (abs‘𝐷) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
41	40	impancom 451	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) → (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
42	41	3ad2ant2 1134	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
43		breq1 5169	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑋 → (𝑧 < (abs‘𝐷) ↔ 𝑋 < (abs‘𝐷)))
44		eleq1 2832	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ 𝑋 ∈ (0...((abs‘𝐷) − 1))))
45	43, 44	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑋 → ((𝑧 < (abs‘𝐷) → 𝑧 ∈ (0...((abs‘𝐷) − 1))) ↔ (𝑋 < (abs‘𝐷) → 𝑋 ∈ (0...((abs‘𝐷) − 1)))))
46	45, 36	vtoclga 3589	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → (𝑋 < (abs‘𝐷) → 𝑋 ∈ (0...((abs‘𝐷) − 1))))
47	46	imp 406	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) → 𝑋 ∈ (0...((abs‘𝐷) − 1)))
48	7, 8	divalglem7 16447	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0...((abs‘𝐷) − 1)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
49	47, 48	sylan 579	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
50	49	3adant2 1131	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
51	50	con2d 134	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → ((𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1)) → ¬ 𝐾 ≠ 0))
52	42, 51	syld 47	. . . . . 6 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → ¬ 𝐾 ≠ 0))
53		df-ne 2947	. . . . . . 7 ⊢ (𝐾 ≠ 0 ↔ ¬ 𝐾 = 0)
54	53	con2bii 357	. . . . . 6 ⊢ (𝐾 = 0 ↔ ¬ 𝐾 ≠ 0)
55	52, 54	imbitrrdi 252	. . . . 5 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → 𝐾 = 0))
56	22, 55	sylbid 240	. . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝐾 = 0))
57		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝐾 = 0 → (𝐾 · (abs‘𝐷)) = (0 · (abs‘𝐷)))
58	10	nncni 12303	. . . . . . . . . . . 12 ⊢ (abs‘𝐷) ∈ ℂ
59	58	mul02i 11479	. . . . . . . . . . 11 ⊢ (0 · (abs‘𝐷)) = 0
60	57, 59	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝐾 = 0 → (𝐾 · (abs‘𝐷)) = 0)
61	60	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝐾 = 0 → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 0 = (𝑌 − 𝑋)))
62	61	biimpac 478	. . . . . . . 8 ⊢ (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 0 = (𝑌 − 𝑋))
63		subeq0 11562	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑌 − 𝑋) = 0 ↔ 𝑌 = 𝑋))
64	5, 6, 63	syl2anr 596	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑌 − 𝑋) = 0 ↔ 𝑌 = 𝑋))
65		eqcom 2747	. . . . . . . . 9 ⊢ ((𝑌 − 𝑋) = 0 ↔ 0 = (𝑌 − 𝑋))
66		eqcom 2747	. . . . . . . . 9 ⊢ (𝑌 = 𝑋 ↔ 𝑋 = 𝑌)
67	64, 65, 66	3bitr3g 313	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (0 = (𝑌 − 𝑋) ↔ 𝑋 = 𝑌))
68	62, 67	imbitrid 244	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 𝑋 = 𝑌))
69	68	ad2ant2r 746	. . . . . 6 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷))) → (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 𝑋 = 𝑌))
70	69	3adant3 1132	. . . . 5 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 𝑋 = 𝑌))
71	70	expd 415	. . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → (𝐾 = 0 → 𝑋 = 𝑌)))
72	56, 71	mpdd 43	. . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌))
73	72	3expia 1121	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌)))
74	73	an4s 659	1 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑋 < (abs‘𝐷) ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 {crab 3443 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ...cfz 13567 abscabs 15283 ∥ cdvds 16302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: divalglem9 16449

W3C validator