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Theorem divalglem8 16346

Description: Lemma for divalg 16349. (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 4041	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ ℕ₀
3		nn0sscn 12423	. . . . . . . . . . . 12 ⊢ ℕ₀ ⊆ ℂ
4	2, 3	sstri 3953	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ℂ
5	4	sseli 3939	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑆 → 𝑌 ∈ ℂ)
6	4	sseli 3939	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ ℂ)
7		divalglem8.2	. . . . . . . . . . . . . 14 ⊢ 𝐷 ∈ ℤ
8		divalglem8.3	. . . . . . . . . . . . . 14 ⊢ 𝐷 ≠ 0
9		nnabscl 15268	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (abs‘𝐷) ∈ ℕ)
10	7, 8, 9	mp2an 692	. . . . . . . . . . . . 13 ⊢ (abs‘𝐷) ∈ ℕ
11	10	nnzi 12533	. . . . . . . . . . . 12 ⊢ (abs‘𝐷) ∈ ℤ
12		zmulcl 12558	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝐾 · (abs‘𝐷)) ∈ ℤ)
13	11, 12	mpan2 691	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℤ)
14	13	zcnd 12615	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 · (abs‘𝐷)) ∈ ℂ)
15		subadd 11400	. . . . . . . . . 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 2736	. . . . . . . 8 ⊢ ((𝑌 − 𝑋) = (𝐾 · (abs‘𝐷)) ↔ (𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋))
19		eqcom 2736	. . . . . . . 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 1178	. . . . . 6 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑋 < (abs‘𝐷)) ∧ 𝑌 ∈ 𝑆 ∧ 𝐾 ∈ ℤ) → ((𝐾 · (abs‘𝐷)) = (𝑌 − 𝑋) ↔ 𝑌 = (𝑋 + (𝐾 · (abs‘𝐷)))))
22	21	3adant2r 1180	. . . . 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 5105	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑌 → (𝑧 < (abs‘𝐷) ↔ 𝑌 < (abs‘𝐷)))
24		eleq1 2816	. . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℕ₀)
27		elnn0z 12518	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ₀ ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧))
28	26, 27	sylib 218	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 < (abs‘𝐷)) → (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷)))
32		0z 12516	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
33		elfzm11 13532	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ (abs‘𝐷) ∈ ℤ) → (𝑧 ∈ (0...((abs‘𝐷) − 1)) ↔ (𝑧 ∈ ℤ ∧ 0 ≤ 𝑧 ∧ 𝑧 < (abs‘𝐷))))
34	32, 11, 33	mp2an 692	. . . . . . . . . . . . 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 3540	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑆 → (𝑌 < (abs‘𝐷) → 𝑌 ∈ (0...((abs‘𝐷) − 1))))
38		eleq1 2816	. . . . . . . . . . 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 5105	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑋 → (𝑧 < (abs‘𝐷) ↔ 𝑋 < (abs‘𝐷)))
44		eleq1 2816	. . . . . . . . . . . . 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 3540	. . . . . . . . . . 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 16345	. . . . . . . . . 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 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 2926	. . . . . . 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 7376	. . . . . . . . . . 11 ⊢ (𝐾 = 0 → (𝐾 · (abs‘𝐷)) = (0 · (abs‘𝐷)))
58	10	nncni 12172	. . . . . . . . . . . 12 ⊢ (abs‘𝐷) ∈ ℂ
59	58	mul02i 11339	. . . . . . . . . . 11 ⊢ (0 · (abs‘𝐷)) = 0
60	57, 59	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝐾 = 0 → (𝐾 · (abs‘𝐷)) = 0)
61	60	eqeq1d 2731	. . . . . . . . 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 11424	. . . . . . . . . 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 2736	. . . . . . . . 9 ⊢ ((𝑌 − 𝑋) = 0 ↔ 0 = (𝑌 − 𝑋))
66		eqcom 2736	. . . . . . . . 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 747	. . . . . 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 660	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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3402 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 ℕcn 12162 ℕ₀cn0 12418 ℤcz 12505 ...cfz 13444 abscabs 15176 ∥ cdvds 16198

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-fz 13445 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178

This theorem is referenced by: divalglem9 16347

W3C validator