Theorem constrextdg2 34267

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 = (ℂfld ↾s 𝑒)
constrextdg2.2	⊢ 𝐹 = (ℂfld ↾s 𝑓)
constrextdg2.l	⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
constrextdg2.n	⊢ (𝜑 → 𝑁 ∈ ω)

Assertion

Ref	Expression
constrextdg2	⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((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 6882	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
3	2	sseq1d 3965	. . . . 5 ⊢ (𝑚 = ∅ → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘𝑣)))
4	3	anbi2d 642	. . . 4 ⊢ (𝑚 = ∅ → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))))
5	4	rexbidv 3188	. . 3 ⊢ (𝑚 = ∅ → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))))
6		fveq2 6882	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
7	6	sseq1d 3965	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑣)))
8	7	anbi2d 642	. . . . 5 ⊢ (𝑚 = 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣))))
9	8	rexbidv 3188	. . . 4 ⊢ (𝑚 = 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣))))
10		fveq1 6881	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
11	10	eqeq1d 2764	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
12		fveq2 6882	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
13	12	sseq2d 3966	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝐶‘𝑛) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))
14	11, 13	anbi12d 644	. . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))))
15	14	cbvrexvw 3243	. . . 4 ⊢ (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)))
16	9, 15	bitrdi 290	. . 3 ⊢ (𝑚 = 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))))
17		fveq2 6882	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
18	17	sseq1d 3965	. . . . 5 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
19	18	anbi2d 642	. . . 4 ⊢ (𝑚 = suc 𝑛 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
20	19	rexbidv 3188	. . 3 ⊢ (𝑚 = suc 𝑛 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
21		fveq2 6882	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
22	21	sseq1d 3965	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑚) ⊆ (lastS‘𝑣) ↔ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
23	22	anbi2d 642	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))))
24	23	rexbidv 3188	. . 3 ⊢ (𝑚 = 𝑁 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣))))
25		fveq1 6881	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (𝑣‘0) = (⟨“ℚ”⟩‘0))
26	25	eqeq1d 2764	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝑣‘0) = ℚ ↔ (⟨“ℚ”⟩‘0) = ℚ))
27		fveq2 6882	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (lastS‘𝑣) = (lastS‘⟨“ℚ”⟩))
28	27	sseq2d 3966	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝐶‘∅) ⊆ (lastS‘𝑣) ↔ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
29	26, 28	anbi12d 644	. . . . 5 ⊢ (𝑣 = ⟨“ℚ”⟩ → (((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)) ↔ ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩))))
30		cndrng 21620	. . . . . . . 8 ⊢ ℂfld ∈ DivRing
31		qsubdrg 21638	. . . . . . . . 9 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
32	31	simpli 489	. . . . . . . 8 ⊢ ℚ ∈ (SubRing‘ℂfld)
33	31	simpri 491	. . . . . . . 8 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
34		issdrg 20960	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
35	30, 32, 33, 34	mpbir3an 1360	. . . . . . 7 ⊢ ℚ ∈ (SubDRing‘ℂfld)
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → ℚ ∈ (SubDRing‘ℂfld))
37	36	s1chn 18714	. . . . 5 ⊢ (⊤ → ⟨“ℚ”⟩ ∈ ( < Chain (SubDRing‘ℂfld)))
38		s1fv 14682	. . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (⟨“ℚ”⟩‘0) = ℚ)
39	36, 38	syl 18	. . . . . 6 ⊢ (⊤ → (⟨“ℚ”⟩‘0) = ℚ)
40		0z 12630	. . . . . . . . 9 ⊢ 0 ∈ ℤ
41		1z 12652	. . . . . . . . 9 ⊢ 1 ∈ ℤ
42		prssi 4785	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
43	40, 41, 42	mp2an 705	. . . . . . . 8 ⊢ {0, 1} ⊆ ℤ
44		zssq 13009	. . . . . . . 8 ⊢ ℤ ⊆ ℚ
45	43, 44	sstri 3943	. . . . . . 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 34255	. . . . . . 7 ⊢ (𝐶‘∅) = {0, 1}
48		lsws1 14683	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (lastS‘⟨“ℚ”⟩) = ℚ)
49	35, 48	ax-mp 5	. . . . . . 7 ⊢ (lastS‘⟨“ℚ”⟩) = ℚ
50	45, 47, 49	3sstr4i 3985	. . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)
51	39, 50	jctir 530	. . . . 5 ⊢ (⊤ → ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
52	29, 37, 51	rspcedvdw 3582	. . . 4 ⊢ (⊤ → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣)))
53	52	mptru 1577	. . 3 ⊢ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘𝑣))
54		constrextdg2.1	. . . . . 6 ⊢ 𝐸 = (ℂfld ↾s 𝑒)
55		constrextdg2.2	. . . . . 6 ⊢ 𝐹 = (ℂfld ↾s 𝑓)
56		constrextdg2.l	. . . . . 6 ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
57		simplll 787	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑛 ∈ ω)
58		simpllr 788	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → 𝑢 ∈ ( < Chain (SubDRing‘ℂfld)))
59		simplr 781	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝑢‘0) = ℚ)
60		simpr 490	. . . . . 6 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → (𝐶‘𝑛) ⊆ (lastS‘𝑢))
61	46, 54, 55, 56, 57, 58, 59, 60	constrextdg2lem 34266	. . . . 5 ⊢ ((((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑢‘0) = ℚ) ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
62	61	anasss 472	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝑢 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢))) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣)))
63	62	rexlimdva2 3167	. . 3 ⊢ (𝑛 ∈ ω → (∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ (𝐶‘𝑛) ⊆ (lastS‘𝑢)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑛) ⊆ (lastS‘𝑣))))
64	5, 16, 20, 24, 53, 63	finds 7897	. 2 ⊢ (𝑁 ∈ ω → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
65	1, 64	syl 18	1 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088  {crab 3414  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  {cpr 4589   class class class wbr 5107  {copab 5171   ↦ cmpt 5190  suc csuc 6363  ‘cfv 6537  (class class class)co 7417  ωcom 7866  reccrdg 8402  ℂcc 11126  ℝcr 11127  0cc0 11128  1c1 11129   + caddc 11131   · cmul 11133   − cmin 11469  2c2 12323  ℤcz 12619  ℚcq 13001  lastSclsw 14631  ⟨“cs1 14666  ∗ccj 15187  ℑcim 15189  abscabs 15325   ↾_s cress 17328   Chain cchn 18699  SubRingcsubrg 20737  DivRingcdr 20896  SubDRingcsdrg 20958  ℂ_fldccnfld 21591  /_FldExtcfldext 34156  [:]cextdg 34158

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-reg 9568  ax-inf2 9624  ax-ac2 10469  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206  ax-addf 11207  ax-mulf 11208

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-ofr 7683  df-rpss 7728  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-sup 9416  df-inf 9417  df-oi 9486  df-r1 9750  df-rank 9751  df-scott 9872  df-dju 9910  df-card 9948  df-acn 9951  df-ac 10123  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-xnn0 12606  df-z 12620  df-dec 12741  df-uz 12892  df-q 13002  df-rp 13047  df-ico 13408  df-fz 13566  df-fzo 13714  df-seq 14070  df-exp 14130  df-hash 14399  df-word 14583  df-lsw 14632  df-concat 14640  df-s1 14667  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-starv 17363  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ocomp 17369  df-ds 17370  df-unif 17371  df-hom 17372  df-cco 17373  df-0g 17532  df-gsum 17533  df-prds 17538  df-pws 17540  df-imas 17600  df-qus 17601  df-mre 17676  df-mrc 17677  df-mri 17678  df-acs 17679  df-proset 18388  df-drs 18389  df-poset 18407  df-ipo 18622  df-chn 18700  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-mhm 18897  df-submnd 18898  df-grp 19066  df-minusg 19067  df-sbg 19068  df-mulg 19197  df-subg 19252  df-nsg 19253  df-eqg 19254  df-ghm 19347  df-gim 19392  df-cntz 19450  df-oppg 19479  df-lsm 19769  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-srg 20332  df-ring 20380  df-cring 20381  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-irred 20506  df-invr 20535  df-dvr 20548  df-rhm 20619  df-nzr 20679  df-subrng 20714  df-subrg 20738  df-rlreg 20862  df-domn 20863  df-idom 20864  df-drng 20898  df-field 20899  df-sdrg 20959  df-lmod 21052  df-lss 21122  df-lsp 21162  df-lmhm 21212  df-lmim 21213  df-lmic 21214  df-lbs 21265  df-lvec 21293  df-sra 21363  df-rgmod 21364  df-lidl 21401  df-rsp 21402  df-2idl 21458  df-lpidl 21559  df-lpir 21560  df-pid 21574  df-cnfld 21592  df-dsmm 21951  df-frlm 21966  df-uvc 22002  df-lindf 22025  df-linds 22026  df-assa 22074  df-asp 22075  df-ascl 22076  df-psr 22130  df-mvr 22131  df-mpl 22132  df-opsr 22134  df-evls 22296  df-evl 22297  df-psr1 22411  df-vr1 22412  df-ply1 22413  df-coe1 22414  df-evls1 22546  df-evl1 22547  df-mdeg 26287  df-deg1 26288  df-mon1 26363  df-uc1p 26364  df-q1p 26365  df-r1p 26366  df-ig1p 26367  df-fldgen 33760  df-mxidl 33871  df-dim 34118  df-fldext 34159  df-extdg 34160  df-irng 34202  df-minply 34218

This theorem is used by:  constrext2chnlem  34268