constrextdg2 - Mathbox for Thierry Arnoux

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Theorem constrextdg2 34000

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 6856	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
3	2	sseq1d 3962	. . . . 5 ⊢ (𝑚 = ∅ → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘𝑣)))
4	3	anbi2d 638	. . . 4 ⊢ (𝑚 = ∅ → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))))
5	4	rexbidv 3180	. . 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 6856	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
7	6	sseq1d 3962	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑣)))
8	7	anbi2d 638	. . . . 5 ⊢ (𝑚 = 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣))))
9	8	rexbidv 3180	. . . 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 6855	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
11	10	eqeq1d 2758	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
12		fveq2 6856	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
13	12	sseq2d 3963	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝐶‘𝑛) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))
14	11, 13	anbi12d 640	. . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))))
15	14	cbvrexvw 3235	. . . 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 289	. . 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 6856	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
18	17	sseq1d 3962	. . . . 5 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
19	18	anbi2d 638	. . . 4 ⊢ (𝑚 = suc 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
20	19	rexbidv 3180	. . 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 6856	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
22	21	sseq1d 3962	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
23	22	anbi2d 638	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))))
24	23	rexbidv 3180	. . 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 6855	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (𝑣‘0) = (⟨“ℚ”⟩‘0))
26	25	eqeq1d 2758	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝑣‘0) = ℚ ↔ (⟨“ℚ”⟩‘0) = ℚ))
27		fveq2 6856	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (lastS‘𝑣) = (lastS‘⟨“ℚ”⟩))
28	27	sseq2d 3963	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝐶‘∅) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
29	26, 28	anbi12d 640	. . . . 5 ⊢ (𝑣 = ⟨“ℚ”⟩ → (((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)) ↔ ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩))))
30		cndrng 21426	. . . . . . . 8 ⊢ ℂ_fld ∈ DivRing
31		qsubdrg 21444	. . . . . . . . 9 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
32	31	simpli 486	. . . . . . . 8 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
33	31	simpri 488	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ DivRing
34		issdrg 20810	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) ↔ (ℂ_fld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing))
35	30, 32, 33, 34	mpbir3an 1351	. . . . . . 7 ⊢ ℚ ∈ (SubDRing‘ℂ_fld)
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → ℚ ∈ (SubDRing‘ℂ_fld))
37	36	s1chn 18628	. . . . 5 ⊢ (⊤ → ⟨“ℚ”⟩ ∈ ( < Chain (SubDRing‘ℂ_fld)))
38		s1fv 14614	. . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) → (⟨“ℚ”⟩‘0) = ℚ)
39	36, 38	syl 17	. . . . . 6 ⊢ (⊤ → (⟨“ℚ”⟩‘0) = ℚ)
40		0z 12569	. . . . . . . . 9 ⊢ 0 ∈ ℤ
41		1z 12591	. . . . . . . . 9 ⊢ 1 ∈ ℤ
42		prssi 4773	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
43	40, 41, 42	mp2an 700	. . . . . . . 8 ⊢ {0, 1} ⊆ ℤ
44		zssq 12947	. . . . . . . 8 ⊢ ℤ ⊆ ℚ
45	43, 44	sstri 3940	. . . . . . 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 33988	. . . . . . 7 ⊢ (𝐶‘∅) = {0, 1}
48		lsws1 14615	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂ_fld) → (lastS‘⟨“ℚ”⟩) = ℚ)
49	35, 48	ax-mp 5	. . . . . . 7 ⊢ (lastS‘⟨“ℚ”⟩) = ℚ
50	45, 47, 49	3sstr4i 3982	. . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)
51	39, 50	jctir 527	. . . . 5 ⊢ (⊤ → ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
52	29, 37, 51	rspcedvdw 3579	. . . 4 ⊢ (⊤ → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂ_fld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)))
53	52	mptru 1561	. . 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 782	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑛 ∈ ω)
58		simpllr 783	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld)))
59		simplr 776	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂ_fld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝑢‘0) = ℚ)
60		simpr 487	. . . . . 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 33999	. . . . 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 469	. . . 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 3159	. . 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 7866	. 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 398 ∨ w3o 1094 ∧ w3a 1095 = wceq 1554 ⊤wtru 1555 ∈ wcel 2136 ≠ wne 2951 ∃wrex 3080 {crab 3408 Vcvv 3448 ⊆ wss 3899 ∅c0 4280 {cpr 4578 class class class wbr 5094 {copab 5156 ↦ cmpt 5175 suc csuc 6337 ‘cfv 6510 (class class class)co 7385 ωcom 7835 reccrdg 8368 ℂcc 11061 ℝcr 11062 0cc0 11063 1c1 11064 + caddc 11066 · cmul 11068 − cmin 11404 2c2 12262 ℤcz 12558 ℚcq 12939 lastSclsw 14565 ⟨“cs1 14599 ∗ccj 15099 ℑcim 15101 abscabs 15237 ↾_s cress 17242 Chain cchn 18613 SubRingcsubrg 20591 DivRingcdr 20751 SubDRingcsdrg 20808 ℂ_fldccnfld 21397 /_FldExtcfldext 33889 [:]cextdg 33891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-reg 9530 ax-inf2 9586 ax-ac2 10410 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141 ax-addf 11142 ax-mulf 11143

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-ofr 7650 df-rpss 7695 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-tpos 8194 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-oadd 8429 df-er 8666 df-ec 8668 df-qs 8672 df-map 8798 df-pm 8799 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-sup 9378 df-inf 9379 df-oi 9448 df-r1 9712 df-rank 9713 df-dju 9849 df-card 9887 df-acn 9890 df-ac 10062 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-xnn0 12545 df-z 12559 df-dec 12679 df-uz 12830 df-q 12940 df-rp 12984 df-ico 13345 df-fz 13503 df-fzo 13650 df-seq 14005 df-exp 14065 df-hash 14334 df-word 14517 df-lsw 14566 df-concat 14574 df-s1 14600 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ocomp 17283 df-ds 17284 df-unif 17285 df-hom 17286 df-cco 17287 df-0g 17446 df-gsum 17447 df-prds 17452 df-pws 17454 df-imas 17514 df-qus 17515 df-mre 17590 df-mrc 17591 df-mri 17592 df-acs 17593 df-proset 18302 df-drs 18303 df-poset 18321 df-ipo 18536 df-chn 18614 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-mhm 18793 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19230 df-gim 19275 df-cntz 19333 df-oppg 19362 df-lsm 19652 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-srg 20209 df-ring 20257 df-cring 20258 df-oppr 20358 df-dvdsr 20378 df-unit 20379 df-irred 20380 df-invr 20409 df-dvr 20422 df-rhm 20493 df-nzr 20535 df-subrng 20568 df-subrg 20592 df-rlreg 20716 df-domn 20717 df-idom 20718 df-drng 20753 df-field 20754 df-sdrg 20809 df-lmod 20902 df-lss 20972 df-lsp 21012 df-lmhm 21062 df-lmim 21063 df-lmic 21064 df-lbs 21115 df-lvec 21143 df-sra 21213 df-rgmod 21214 df-lidl 21251 df-rsp 21252 df-2idl 21293 df-lpidl 21365 df-lpir 21366 df-pid 21380 df-cnfld 21398 df-dsmm 21757 df-frlm 21772 df-uvc 21808 df-lindf 21831 df-linds 21832 df-assa 21878 df-asp 21879 df-ascl 21880 df-psr 21934 df-mvr 21935 df-mpl 21936 df-opsr 21938 df-evls 22100 df-evl 22101 df-psr1 22215 df-vr1 22216 df-ply1 22217 df-coe1 22218 df-evls1 22351 df-evl1 22352 df-mdeg 26088 df-deg1 26089 df-mon1 26164 df-uc1p 26165 df-q1p 26166 df-r1p 26167 df-ig1p 26168 df-fldgen 33452 df-mxidl 33602 df-dim 33851 df-fldext 33892 df-extdg 33893 df-irng 33935 df-minply 33951

This theorem is referenced by: constrext2chnlem 34001

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