constrextdg2 - Mathbox for Thierry Arnoux

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Theorem constrextdg2 33926

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 6842	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
3	2	sseq1d 3967	. . . . 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 3162	. . 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 6842	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
7	6	sseq1d 3967	. . . . . 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 3162	. . . 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 6841	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
11	10	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
12		fveq2 6842	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
13	12	sseq2d 3968	. . . . . 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 3217	. . . 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 6842	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
18	17	sseq1d 3967	. . . . 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 3162	. . 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 6842	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
22	21	sseq1d 3967	. . . . 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 3162	. . 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 6841	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (𝑣‘0) = (⟨“ℚ”⟩‘0))
26	25	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝑣‘0) = ℚ ↔ (⟨“ℚ”⟩‘0) = ℚ))
27		fveq2 6842	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (lastS‘𝑣) = (lastS‘⟨“ℚ”⟩))
28	27	sseq2d 3968	. . . . . 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 21365	. . . . . . . 8 ⊢ ℂ_fld ∈ DivRing
31		qsubdrg 21386	. . . . . . . . 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 20733	. . . . . . . 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 18555	. . . . 5 ⊢ (⊤ → ⟨“ℚ”⟩ ∈ ( < Chain (SubDRing‘ℂ_fld)))
38		s1fv 14546	. . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) → (⟨“ℚ”⟩‘0) = ℚ)
39	36, 38	syl 17	. . . . . 6 ⊢ (⊤ → (⟨“ℚ”⟩‘0) = ℚ)
40		0z 12511	. . . . . . . . 9 ⊢ 0 ∈ ℤ
41		1z 12533	. . . . . . . . 9 ⊢ 1 ∈ ℤ
42		prssi 4779	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
43	40, 41, 42	mp2an 693	. . . . . . . 8 ⊢ {0, 1} ⊆ ℤ
44		zssq 12881	. . . . . . . 8 ⊢ ℤ ⊆ ℚ
45	43, 44	sstri 3945	. . . . . . 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 33914	. . . . . . 7 ⊢ (𝐶‘∅) = {0, 1}
48		lsws1 14547	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) → (lastS‘⟨“ℚ”⟩) = ℚ)
49	35, 48	ax-mp 5	. . . . . . 7 ⊢ (lastS‘⟨“ℚ”⟩) = ℚ
50	45, 47, 49	3sstr4i 3987	. . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)
51	39, 50	jctir 520	. . . . 5 ⊢ (⊤ → ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
52	29, 37, 51	rspcedvdw 3581	. . . 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 33925	. . . . 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 3141	. . 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 7848	. 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 3062 {crab 3401 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 {cpr 4584 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 suc csuc 6327 ‘cfv 6500 (class class class)co 7368 ωcom 7818 reccrdg 8350 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11376 2c2 12212 ℤcz 12500 ℚcq 12873 lastSclsw 14497 ⟨“cs1 14531 ∗ccj 15031 ℑcim 15033 abscabs 15169 ↾_s cress 17169 Chain cchn 18540 SubRingcsubrg 20514 DivRingcdr 20674 SubDRingcsdrg 20731 ℂ_fldccnfld 21321 /_FldExtcfldext 33815 [:]cextdg 33817

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-reg 9509 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

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 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-rpss 7678 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-inf 9358 df-oi 9427 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-ico 13279 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-word 14449 df-lsw 14498 df-concat 14506 df-s1 14532 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ocomp 17210 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-imas 17441 df-qus 17442 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18229 df-drs 18230 df-poset 18248 df-ipo 18463 df-chn 18541 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-gim 19200 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-irred 20307 df-invr 20336 df-dvr 20349 df-rhm 20420 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-idom 20641 df-drng 20676 df-field 20677 df-sdrg 20732 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lmim 20987 df-lmic 20988 df-lbs 21039 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-lidl 21175 df-rsp 21176 df-2idl 21217 df-lpidl 21289 df-lpir 21290 df-pid 21304 df-cnfld 21322 df-dsmm 21699 df-frlm 21714 df-uvc 21750 df-lindf 21773 df-linds 21774 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evls1 22271 df-evl1 22272 df-mdeg 26028 df-deg1 26029 df-mon1 26104 df-uc1p 26105 df-q1p 26106 df-r1p 26107 df-ig1p 26108 df-fldgen 33404 df-mxidl 33552 df-dim 33776 df-fldext 33818 df-extdg 33819 df-irng 33861 df-minply 33877

This theorem is referenced by: constrext2chnlem 33927

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