Theorem constrextdg2 34315

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 6873	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
3	2	sseq1d 3961	. . . . 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 3186	. . 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 6873	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
7	6	sseq1d 3961	. . . . . 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 3186	. . . 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 6872	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
11	10	eqeq1d 2762	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
12		fveq2 6873	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
13	12	sseq2d 3962	. . . . . 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 3241	. . . 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 6873	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
18	17	sseq1d 3961	. . . . 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 3186	. . 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 6873	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
22	21	sseq1d 3961	. . . . 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 3186	. . 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 6872	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (𝑣‘0) = (⟨“ℚ”⟩‘0))
26	25	eqeq1d 2762	. . . . . 6 ⊢ (𝑣 = ⟨“ℚ”⟩ → ((𝑣‘0) = ℚ ↔ (⟨“ℚ”⟩‘0) = ℚ))
27		fveq2 6873	. . . . . . 7 ⊢ (𝑣 = ⟨“ℚ”⟩ → (lastS‘𝑣) = (lastS‘⟨“ℚ”⟩))
28	27	sseq2d 3962	. . . . . 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 21669	. . . . . . . 8 ⊢ ℂfld ∈ DivRing
31		qsubdrg 21687	. . . . . . . . 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 21007	. . . . . . . 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 18756	. . . . 5 ⊢ (⊤ → ⟨“ℚ”⟩ ∈ ( < Chain (SubDRing‘ℂfld)))
38		s1fv 14726	. . . . . . 7 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (⟨“ℚ”⟩‘0) = ℚ)
39	36, 38	syl 18	. . . . . 6 ⊢ (⊤ → (⟨“ℚ”⟩‘0) = ℚ)
40		0z 12674	. . . . . . . . 9 ⊢ 0 ∈ ℤ
41		1z 12696	. . . . . . . . 9 ⊢ 1 ∈ ℤ
42		prssi 4781	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ⊆ ℤ)
43	40, 41, 42	mp2an 705	. . . . . . . 8 ⊢ {0, 1} ⊆ ℤ
44		zssq 13053	. . . . . . . 8 ⊢ ℤ ⊆ ℚ
45	43, 44	sstri 3939	. . . . . . 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 34303	. . . . . . 7 ⊢ (𝐶‘∅) = {0, 1}
48		lsws1 14727	. . . . . . . 8 ⊢ (ℚ ∈ (SubDRing‘ℂfld) → (lastS‘⟨“ℚ”⟩) = ℚ)
49	35, 48	ax-mp 5	. . . . . . 7 ⊢ (lastS‘⟨“ℚ”⟩) = ℚ
50	45, 47, 49	3sstr4i 3981	. . . . . 6 ⊢ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)
51	39, 50	jctir 530	. . . . 5 ⊢ (⊤ → ((⟨“ℚ”⟩‘0) = ℚ ∧ (𝐶‘∅) ⊆ (lastS‘⟨“ℚ”⟩)))
52	29, 37, 51	rspcedvdw 3579	. . . 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 34314	. . . . 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 3165	. . 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 7891	. 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 2955  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  {cpr 4585   class class class wbr 5102  {copab 5166   ↦ cmpt 5185  suc csuc 6353  ‘cfv 6527  (class class class)co 7408  ωcom 7860  reccrdg 8395  ℂcc 11170  ℝcr 11171  0cc0 11172  1c1 11173   + caddc 11175   · cmul 11177   − cmin 11513  2c2 12367  ℤcz 12663  ℚcq 13045  lastSclsw 14675  ⟨“cs1 14710  ∗ccj 15231  ℑcim 15233  abscabs 15369   ↾_s cress 17370   Chain cchn 18741  SubRingcsubrg 20783  DivRingcdr 20942  SubDRingcsdrg 21005  ℂ_fldccnfld 21640  /_FldExtcfldext 34204  [:]cextdg 34206

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-reg 9564  ax-inf2 9620  ax-ac2 10513  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-addrcl 11233  ax-mulcl 11234  ax-mulrcl 11235  ax-mulcom 11236  ax-addass 11237  ax-mulass 11238  ax-distr 11239  ax-i2m1 11240  ax-1ne0 11241  ax-1rid 11242  ax-rnegex 11243  ax-rrecex 11244  ax-cnre 11245  ax-pre-lttri 11246  ax-pre-lttrn 11247  ax-pre-ltadd 11248  ax-pre-mulgt0 11249  ax-pre-sup 11250  ax-addf 11251  ax-mulf 11252

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7676  df-ofr 7677  df-rpss 7722  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-ec 8697  df-qs 8701  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-sup 9412  df-inf 9413  df-oi 9482  df-r1 9746  df-rank 9747  df-scott 9901  df-dju 9954  df-card 9992  df-acn 9995  df-ac 10167  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-le 11321  df-sub 11515  df-neg 11516  df-div 11944  df-nn 12306  df-2 12375  df-3 12376  df-4 12377  df-5 12378  df-6 12379  df-7 12380  df-8 12381  df-9 12382  df-n0 12577  df-xnn0 12650  df-z 12664  df-dec 12785  df-uz 12936  df-q 13046  df-rp 13091  df-ico 13452  df-fz 13610  df-fzo 13758  df-seq 14114  df-exp 14174  df-hash 14443  df-word 14627  df-lsw 14676  df-concat 14684  df-s1 14711  df-cj 15234  df-re 15235  df-im 15236  df-sqrt 15370  df-abs 15371  df-struct 17287  df-sets 17304  df-slot 17322  df-ndx 17334  df-base 17350  df-ress 17371  df-plusg 17403  df-mulr 17404  df-starv 17405  df-sca 17406  df-vsca 17407  df-ip 17408  df-tset 17409  df-ple 17410  df-ocomp 17411  df-ds 17412  df-unif 17413  df-hom 17414  df-cco 17415  df-0g 17574  df-gsum 17575  df-prds 17580  df-pws 17582  df-imas 17642  df-qus 17643  df-mre 17718  df-mrc 17719  df-mri 17720  df-acs 17721  df-proset 18430  df-drs 18431  df-poset 18449  df-ipo 18664  df-chn 18742  df-mgm 18778  df-sgrp 18870  df-mnd 18886  df-mhm 18940  df-submnd 18941  df-grp 19109  df-minusg 19110  df-sbg 19111  df-mulg 19240  df-subg 19295  df-nsg 19296  df-eqg 19297  df-ghm 19390  df-gim 19435  df-cntz 19493  df-oppg 19522  df-lsm 19812  df-cmn 19958  df-abl 19959  df-mgp 20323  df-rng 20337  df-ur 20370  df-srg 20375  df-ring 20423  df-cring 20424  df-oppr 20529  df-dvdsr 20549  df-unit 20550  df-irred 20551  df-invr 20580  df-dvr 20593  df-rhm 20664  df-nzr 20725  df-subrng 20760  df-subrg 20784  df-rlreg 20908  df-domn 20909  df-idom 20910  df-drng 20944  df-field 20945  df-sdrg 21006  df-lmod 21099  df-lss 21169  df-lsp 21209  df-lmhm 21259  df-lmim 21260  df-lmic 21261  df-lbs 21312  df-lvec 21340  df-sra 21410  df-rgmod 21411  df-lidl 21448  df-rsp 21449  df-2idl 21505  df-lpidl 21608  df-lpir 21609  df-pid 21623  df-cnfld 21641  df-dsmm 22000  df-frlm 22015  df-uvc 22051  df-lindf 22074  df-linds 22075  df-assa 22123  df-asp 22124  df-ascl 22125  df-psr 22179  df-mvr 22180  df-mpl 22181  df-opsr 22183  df-evls 22345  df-evl 22346  df-psr1 22460  df-vr1 22461  df-ply1 22462  df-coe1 22463  df-evls1 22595  df-evl1 22596  df-mdeg 26335  df-deg1 26336  df-mon1 26411  df-uc1p 26412  df-q1p 26413  df-r1p 26414  df-ig1p 26415  df-fldgen 33807  df-mxidl 33919  df-dim 34166  df-fldext 34207  df-extdg 34208  df-irng 34250  df-minply 34266

This theorem is used by:  constrext2chnlem  34316