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Theorem divalglem8 16109

Description: Lemma for divalg 16112. (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 4015	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ ℕ₀
3		nn0sscn 12238	. . . . . . . . . . . 12 ⊢ ℕ₀ ⊆ ℂ
4	2, 3	sstri 3930	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ℂ
5	4	sseli 3917	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑆 → 𝑌 ∈ ℂ)
6	4	sseli 3917	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ ℂ)
7		divalglem8.2	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ ℤ
8		divalglem8.3	. . . . . . . . . . . . . 14 ⊢ 𝐷 ≠ 0
9		nnabscl 15037	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ)
10	7, 8, 9	mp2an 689	. . . . . . . . . . . . 13 ⊢ (abs‘𝐷) ∈ ℕ
11	10	nnzi 12344	. . . . . . . . . . . 12 ⊢ (abs‘𝐷) ∈ ℤ
12		zmulcl 12369	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐾 · (abs‘𝐷)) ∈ ℤ)
13	11, 12	mpan2 688	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℤ)
14	13	zcnd 12427	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℂ)
15		subadd 11224	. . . . . . . . . 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 1159	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝑆 ∧ 𝑋 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) = 𝑌))
17	16	3com12 1122	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) = 𝑌))
18		eqcom 2745	. . . . . . . 8 ⊢ ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋))
19		eqcom 2745	. . . . . . . 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 1176	. . . . . 6 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ 𝑌 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))))
22	21	3adant2r 1178	. . . . 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 5077	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑌 → (𝑧 < (abs‘𝐷) ↔ 𝑌 < (abs‘𝐷)))
24		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑌 → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ 𝑌 ∈ (0...((abs‘𝐷) − 1))))
25	23, 24	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑌 → ((𝑧 < (abs‘𝐷) → 𝑧 ∈ (0...((abs‘𝐷) − 1))) ↔ (𝑌 < (abs‘𝐷) → 𝑌 ∈ (0...((abs‘𝐷) − 1)))))
26	2	sseli 3917	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℕ₀)
27		elnn0z 12332	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ₀ ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧))
28	26, 27	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧))
29	28	anim1i 615	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → ((𝑧 ∈ ℤ ∧ 0 ≤ 𝑧) ∧ 𝑧 < (abs‘𝐷)))
30		df-3an 1088	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)) ↔ ((𝑧 ∈ ℤ ∧ 0 ≤ 𝑧) ∧ 𝑧 < (abs‘𝐷)))
31	29, 30	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)))
32		0z 12330	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
33		elfzm11 13327	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷))))
34	32, 11, 33	mp2an 689	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)))
35	31, 34	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → 𝑧 ∈ (0...((abs‘𝐷) − 1)))
36	35	ex 413	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑆 → (𝑧 < (abs‘𝐷) → 𝑧 ∈ (0...((abs‘𝐷) − 1))))
37	25, 36	vtoclga 3513	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑆 → (𝑌 < (abs‘𝐷) → 𝑌 ∈ (0...((abs‘𝐷) − 1))))
38		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑌 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
39	38	biimpd 228	. . . . . . . . . 10 ⊢ (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑌 ∈ (0...((abs‘𝐷) − 1)) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
40	37, 39	sylan9 508	. . . . . . . . 9 ⊢ ((𝑌 ∈ 𝑆 ∧ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))) → (𝑌 < (abs‘𝐷) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
41	40	impancom 452	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) → (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
42	41	3ad2ant2 1133	. . . . . . 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 5077	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑋 → (𝑧 < (abs‘𝐷) ↔ 𝑋 < (abs‘𝐷)))
44		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ 𝑋 ∈ (0...((abs‘𝐷) − 1))))
45	43, 44	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑋 → ((𝑧 < (abs‘𝐷) → 𝑧 ∈ (0...((abs‘𝐷) − 1))) ↔ (𝑋 < (abs‘𝐷) → 𝑋 ∈ (0...((abs‘𝐷) − 1)))))
46	45, 36	vtoclga 3513	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → (𝑋 < (abs‘𝐷) → 𝑋 ∈ (0...((abs‘𝐷) − 1))))
47	46	imp 407	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) → 𝑋 ∈ (0...((abs‘𝐷) − 1)))
48	7, 8	divalglem7 16108	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (0...((abs‘𝐷) − 1)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
49	47, 48	sylan 580	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝐾 ≠ 0 → ¬ (𝑋 + (𝐾 · (abs‘𝐷))) ∈ (0...((abs‘𝐷) − 1))))
50	49	3adant2 1130	. . . . . . . 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 2944	. . . . . . 7 ⊢ (𝐾 ≠ 0 ↔ ¬ 𝐾 = 0)
54	53	con2bii 358	. . . . . 6 ⊢ (𝐾 = 0 ↔ ¬ 𝐾 ≠ 0)
55	52, 54	syl6ibr 251	. . . . 5 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → (𝑌 = (𝑋 + (𝐾 · (abs‘𝐷))) → 𝐾 = 0))
56	22, 55	sylbid 239	. . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷)) ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝐾 = 0))
57		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝐾 = 0 → (𝐾 · (abs‘𝐷)) = (0 · (abs‘𝐷)))
58	10	nncni 11983	. . . . . . . . . . . 12 ⊢ (abs‘𝐷) ∈ ℂ
59	58	mul02i 11164	. . . . . . . . . . 11 ⊢ (0 · (abs‘𝐷)) = 0
60	57, 59	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝐾 = 0 → (𝐾 · (abs‘𝐷)) = 0)
61	60	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝐾 = 0 → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 0 = (𝑌 − 𝑋)))
62	61	biimpac 479	. . . . . . . 8 ⊢ (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 0 = (𝑌 − 𝑋))
63		subeq0 11247	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑌 − 𝑋) = 0 ↔ 𝑌 = 𝑋))
64	5, 6, 63	syl2anr 597	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑌 − 𝑋) = 0 ↔ 𝑌 = 𝑋))
65		eqcom 2745	. . . . . . . . 9 ⊢ ((𝑌 − 𝑋) = 0 ↔ 0 = (𝑌 − 𝑋))
66		eqcom 2745	. . . . . . . . 9 ⊢ (𝑌 = 𝑋 ↔ 𝑋 = 𝑌)
67	64, 65, 66	3bitr3g 313	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (0 = (𝑌 − 𝑋) ↔ 𝑋 = 𝑌))
68	62, 67	syl5ib 243	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 𝑋 = 𝑌))
69	68	ad2ant2r 744	. . . . . 6 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷))) → (((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ∧ 𝐾 = 0) → 𝑋 = 𝑌))
70	69	3adant3 1131	. . . . 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 416	. . . 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 1120	. 2 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ (𝑌 ∈ 𝑆 ∧ 𝑌 < (abs‘𝐷))) → (𝐾 ∈ ℤ → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) → 𝑋 = 𝑌)))
74	73	an4s 657	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {crab 3068 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ...cfz 13239 abscabs 14945 ∥ cdvds 15963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: divalglem9 16110

W3C validator