constrextdg2 - Mathbox for Thierry Arnoux

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Theorem constrextdg2 33887

Description: Any step (𝐶‘𝑁) of the construction of constructible numbers is contained in the last field of a tower of quadratic field extensions starting with ℚ. See Theorem 7.11 of [Stewart] p. 97. (Contributed by Thierry Arnoux, 19-Oct-2025.)

**Hypotheses**
Ref	Expression
constr0.1	⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
constrextdg2.1	⊢ 𝐸 = (ℂ_fld ↾_s 𝑒)
constrextdg2.2	⊢ 𝐹 = (ℂ_fld ↾_s 𝑓)
constrextdg2.l	⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
constrextdg2.n	⊢ (𝜑 → 𝑁 ∈ ω)

**Assertion**
Ref	Expression
constrextdg2	⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))

Distinct variable groups: < ,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑠,𝑡,𝑣,𝑥 𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑠,𝑡,𝑣,𝑥 𝑡,𝑁,𝑣 Allowed substitution hints: 𝜑(𝑥,𝑣,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑) 𝐸(𝑥,𝑣,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑) 𝐹(𝑥,𝑣,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑) 𝑁(𝑥,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem constrextdg2 Dummy variables 𝑛 𝑢 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		constrextdg2.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ω)
2		fveq2 6835	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
3	2	sseq1d 3966	. . . . 5 ⊢ (𝑚 = ∅ → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘𝑣)))
4	3	anbi2d 631	. . . 4 ⊢ (𝑚 = ∅ → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))))
5	4	rexbidv 3161	. . 3 ⊢ (𝑚 = ∅ → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))))
6		fveq2 6835	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
7	6	sseq1d 3966	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑣)))
8	7	anbi2d 631	. . . . 5 ⊢ (𝑚 = 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣))))
9	8	rexbidv 3161	. . . 4 ⊢ (𝑚 = 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣))))
10		fveq1 6834	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
11	10	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
12		fveq2 6835	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
13	12	sseq2d 3967	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝐶‘𝑛) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))
14	11, 13	anbi12d 633	. . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))))
15	14	cbvrexvw 3216	. . . 4 ⊢ (∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))
16	9, 15	bitrdi 287	. . 3 ⊢ (𝑚 = 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))))
17		fveq2 6835	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
18	17	sseq1d 3966	. . . . 5 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
19	18	anbi2d 631	. . . 4 ⊢ (𝑚 = suc 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
20	19	rexbidv 3161	. . 3 ⊢ (𝑚 = suc 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
21		fveq2 6835	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
22	21	sseq1d 3966	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
23	22	anbi2d 631	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))))
24	23	rexbidv 3161	. . 3 ⊢ (𝑚 = 𝑁 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))))
25		fveq1 6834	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (𝑣‘0) = (⟨“ℚ”⟩‘0))
26	25	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝑣‘0) = ℚ ↔ (⟨“ℚ”⟩‘0) = ℚ))
27		fveq2 6835	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (lastS‘𝑣) = (lastS‘⟨“ℚ”⟩))
28	27	sseq2d 3967	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝐶‘∅) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
29	26, 28	anbi12d 633	. . . . 5 ⊢ (𝑣 = ⟨“ℚ”⟩ → (((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)) ↔ ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩))))
30		cndrng 21357	. . . . . . . 8 ⊢ ℂ_fld ∈ DivRing
31		qsubdrg 21378	. . . . . . . . 9 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
32	31	simpli 483	. . . . . . . 8 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
33	31	simpri 485	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ DivRing
34		issdrg 20725	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) ↔ (ℂ_fld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing))
35	30, 32, 33, 34	mpbir3an 1343	. . . . . . 7 ⊢ ℚ ∈ (SubDRing‘ℂ_fld)
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → ℚ ∈ (SubDRing‘ℂ_fld))
37	36	s1chn 18547	. . . . 5 ⊢ (⊤ → ⟨“ℚ”⟩ ∈ ( < Chain (SubDRing‘ℂ_fld)))
38		s1fv 14538	. . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) → (⟨“ℚ”⟩‘0) = ℚ)
39	36, 38	syl 17	. . . . . 6 ⊢ (⊤ → (⟨“ℚ”⟩‘0) = ℚ)
40		0z 12503	. . . . . . . . 9 ⊢ 0 ∈ ℤ
41		1z 12525	. . . . . . . . 9 ⊢ 1 ∈ ℤ
42		prssi 4778	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
43	40, 41, 42	mp2an 693	. . . . . . . 8 ⊢ {0, 1} ⊆ ℤ
44		zssq 12873	. . . . . . . 8 ⊢ ℤ ⊆ ℚ
45	43, 44	sstri 3944	. . . . . . 7 ⊢ {0, 1} ⊆ ℚ
46		constr0.1	. . . . . . . 8 ⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
47	46	constr0 33875	. . . . . . 7 ⊢ (𝐶‘∅) = {0, 1}
48		lsws1 14539	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) → (lastS‘⟨“ℚ”⟩) = ℚ)
49	35, 48	ax-mp 5	. . . . . . 7 ⊢ (lastS‘⟨“ℚ”⟩) = ℚ
50	45, 47, 49	3sstr4i 3986	. . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)
51	39, 50	jctir 520	. . . . 5 ⊢ (⊤ → ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
52	29, 37, 51	rspcedvdw 3580	. . . 4 ⊢ (⊤ → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)))
53	52	mptru 1549	. . 3 ⊢ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))
54		constrextdg2.1	. . . . . 6 ⊢ 𝐸 = (ℂ_fld ↾_s 𝑒)
55		constrextdg2.2	. . . . . 6 ⊢ 𝐹 = (ℂ_fld ↾_s 𝑓)
56		constrextdg2.l	. . . . . 6 ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
57		simplll 775	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑛 ∈ ω)
58		simpllr 776	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld)))
59		simplr 769	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝑢‘0) = ℚ)
60		simpr 484	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝐶‘𝑛) ⊆ (lastS‘𝑢))
61	46, 54, 55, 56, 57, 58, 59, 60	constrextdg2lem 33886	. . . . 5 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
62	61	anasss 466	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
63	62	rexlimdva2 3140	. . 3 ⊢ (𝑛 ∈ ω → (∃𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
64	5, 16, 20, 24, 53, 63	finds 7840	. 2 ⊢ (𝑁 ∈ ω → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
65	1, 64	syl 17	1 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3061 {crab 3400 Vcvv 3441 ⊆ wss 3902 ∅c0 4286 {cpr 4583 class class class wbr 5099 {copab 5161 ↦ cmpt 5180 suc csuc 6320 ‘cfv 6493 (class class class)co 7360 ωcom 7810 reccrdg 8342 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 − cmin 11368 2c2 12204 ℤcz 12492 ℚcq 12865 lastSclsw 14489 ⟨“cs1 14523 ∗ccj 15023 ℑcim 15025 abscabs 15161 ↾_s cress 17161 Chain cchn 18532 SubRingcsubrg 20506 DivRingcdr 20666 SubDRingcsdrg 20723 ℂ_fldccnfld 21313 /_FldExtcfldext 33776 [:]cextdg 33778

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-reg 9501 ax-inf2 9554 ax-ac2 10377 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-rpss 7670 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-r1 9680 df-rank 9681 df-dju 9817 df-card 9855 df-acn 9858 df-ac 10030 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-ico 13271 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-hash 14258 df-word 14441 df-lsw 14490 df-concat 14498 df-s1 14524 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ocomp 17202 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17365 df-gsum 17366 df-prds 17371 df-pws 17373 df-imas 17433 df-qus 17434 df-mre 17509 df-mrc 17510 df-mri 17511 df-acs 17512 df-proset 18221 df-drs 18222 df-poset 18240 df-ipo 18455 df-chn 18533 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-nsg 19058 df-eqg 19059 df-ghm 19146 df-gim 19192 df-cntz 19250 df-oppg 19279 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-irred 20299 df-invr 20328 df-dvr 20341 df-rhm 20412 df-nzr 20450 df-subrng 20483 df-subrg 20507 df-rlreg 20631 df-domn 20632 df-idom 20633 df-drng 20668 df-field 20669 df-sdrg 20724 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lmhm 20978 df-lmim 20979 df-lmic 20980 df-lbs 21031 df-lvec 21059 df-sra 21129 df-rgmod 21130 df-lidl 21167 df-rsp 21168 df-2idl 21209 df-lpidl 21281 df-lpir 21282 df-pid 21296 df-cnfld 21314 df-dsmm 21691 df-frlm 21706 df-uvc 21742 df-lindf 21765 df-linds 21766 df-assa 21812 df-asp 21813 df-ascl 21814 df-psr 21869 df-mvr 21870 df-mpl 21871 df-opsr 21873 df-evls 22033 df-evl 22034 df-psr1 22124 df-vr1 22125 df-ply1 22126 df-coe1 22127 df-evls1 22263 df-evl1 22264 df-mdeg 26020 df-deg1 26021 df-mon1 26096 df-uc1p 26097 df-q1p 26098 df-r1p 26099 df-ig1p 26100 df-fldgen 33374 df-mxidl 33522 df-dim 33737 df-fldext 33779 df-extdg 33780 df-irng 33822 df-minply 33838

This theorem is referenced by: constrext2chnlem 33888

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