Theorem constrext2chnlem 34057

Description: Lemma for constrext2chn 34066. (Contributed by Thierry Arnoux, 26-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)
constrext2chnlem.q	⊢ 𝑄 = (ℂfld ↾s ℚ)
constrext2chnlem.l	⊢ 𝐿 = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
constrext2chnlem.a	⊢ (𝜑 → 𝐴 ∈ Constr)

Assertion

Ref	Expression
constrext2chnlem	⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))

Distinct variable groups:   < ,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑛,𝑟,𝑠,𝑡,𝑥   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑛,𝑟,𝑠,𝑡,𝑥   𝑡,𝑁   𝐴,𝑛   𝑛,𝐿   𝑄,𝑛   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝑄(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐸(𝑥,𝑡,𝑒,𝑓,𝑛,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑥,𝑡,𝑒,𝑓,𝑛,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐿(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝑁(𝑥,𝑒,𝑓,𝑛,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem constrext2chnlem

Dummy variables 𝑣 𝑚 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2prm 16740	. . . . . 6 ⊢ 2 ∈ ℙ
2	1	a1i 11	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → 2 ∈ ℙ)
3		constrext2chnlem.l	. . . . . . 7 ⊢ 𝐿 = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
4		constrext2chnlem.q	. . . . . . 7 ⊢ 𝑄 = (ℂfld ↾s ℚ)
5	3, 4	oveq12i 7412	. . . . . 6 ⊢ (𝐿[:]𝑄) = ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))
6		cnfldbas 21486	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
7		eqid 2765	. . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ)
8		eqid 2765	. . . . . . . . . 10 ⊢ (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
9		cnfldfld 33577	. . . . . . . . . . 11 ⊢ ℂfld ∈ Field
10	9	a1i 11	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℂfld ∈ Field)
11		cndrng 21511	. . . . . . . . . . . 12 ⊢ ℂfld ∈ DivRing
12		qsubdrg 21529	. . . . . . . . . . . . 13 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
13	12	simpli 488	. . . . . . . . . . . 12 ⊢ ℚ ∈ (SubRing‘ℂfld)
14	12	simpri 490	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
15		issdrg 20860	. . . . . . . . . . . 12 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
16	11, 13, 14, 15	mpbir3an 1358	. . . . . . . . . . 11 ⊢ ℚ ∈ (SubDRing‘ℂfld)
17	16	a1i 11	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℚ ∈ (SubDRing‘ℂfld))
18		constr0.1	. . . . . . . . . . . . . 14 ⊢ 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})
19		nnon 7856	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ω → 𝑚 ∈ On)
20	19	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ On)
21	18, 20	constrsscn 34047	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐶‘𝑚) ⊆ ℂ)
22	21	sselda 3939	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝐴 ∈ ℂ)
23	22	snssd 4748	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → {𝐴} ⊆ ℂ)
24	23	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → {𝐴} ⊆ ℂ)
25	6, 7, 8, 10, 17, 24	fldgenfldext 33975	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))/FldExt(ℂfld ↾s ℚ))
26	25	ad2antrr 738	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))/FldExt(ℂfld ↾s ℚ))
27		extdgcl 33963	. . . . . . . 8 ⊢ ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))/FldExt(ℂfld ↾s ℚ) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℕ0*)
28	26, 27	syl 18	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℕ0*)
29		simpr 489	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝))
30		2z 12617	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
31	30	a1i 11	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → 2 ∈ ℤ)
32		simplr 780	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → 𝑝 ∈ ℕ0)
33	31, 32	zexpcld 14114	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (2↑𝑝) ∈ ℤ)
34	29, 33	eqeltrd 2865	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) ∈ ℤ)
35	34	zred 12691	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) ∈ ℝ)
36		xnn0xr 12573	. . . . . . . . 9 ⊢ (((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℕ0* → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℝ*)
37	26, 27, 36	3syl 19	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℝ*)
38		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Base‘(ℂfld ↾s (lastS‘𝑣))) = (Base‘(ℂfld ↾s (lastS‘𝑣)))
39		constrextdg2.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (ℂfld ↾s 𝑒)
40		constrextdg2.2	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (ℂfld ↾s 𝑓)
41		constrextdg2.l	. . . . . . . . . . . . . . . 16 ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
42		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ ( < Chain (SubDRing‘ℂfld)))
43		simprl 782	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (𝑣‘0) = ℚ)
44	43	oveq2d 7416	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (𝑣‘0)) = (ℂfld ↾s ℚ))
45		eqidd 2766	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣)))
46		simpr 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → 𝑣 = ∅)
47	46	fveq1d 6873	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = (∅‘0))
48		0fv 6912	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅‘0) = ∅
49	48	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (∅‘0) = ∅)
50	47, 49	eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = ∅)
51	43	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = ℚ)
52		1nn 12235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
53		nnq 12977	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℚ
55	54	ne0ii 4299	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℚ ≠ ∅
56	55	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → ℚ ≠ ∅)
57	51, 56	eqnetrd 3027	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) ≠ ∅)
58	57	neneqd 2965	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → ¬ (𝑣‘0) = ∅)
59	50, 58	pm2.65da 828	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ¬ 𝑣 = ∅)
60	59	neqned 2967	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ≠ ∅)
61	42, 60	hashne0 33067	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 0 < (♯‘𝑣))
62	39, 40, 41, 42, 10, 44, 45, 61	fldext2chn 34035	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ((ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s ℚ) ∧ ∃𝑝 ∈ ℕ0 ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)))
63	62	simpld 499	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s ℚ))
64		fldextfld1 33954	. . . . . . . . . . . . . 14 ⊢ ((ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s ℚ) → (ℂfld ↾s (lastS‘𝑣)) ∈ Field)
65	63, 64	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (lastS‘𝑣)) ∈ Field)
66	42	chnwrd 18654	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ Word (SubDRing‘ℂfld))
67		lswcl 14595	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ 𝑣 ≠ ∅) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
68	66, 60, 67	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
69	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℂfld ∈ DivRing)
70		qsscn 12975	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ⊆ ℂ
71	70	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ℚ ⊆ ℂ)
72	71, 23	unssd 4147	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → (ℚ ∪ {𝐴}) ⊆ ℂ)
73	72	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℚ ∪ {𝐴}) ⊆ ℂ)
74	6, 69, 73	fldgensdrg 33550	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld fldGen (ℚ ∪ {𝐴})) ∈ (SubDRing‘ℂfld))
75	7	qrngbas 27741	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ = (Base‘(ℂfld ↾s ℚ))
76	75, 63	fldextsdrg 33961	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℚ ∈ (SubDRing‘(ℂfld ↾s (lastS‘𝑣))))
77	38	sdrgss 20865	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubDRing‘(ℂfld ↾s (lastS‘𝑣))) → ℚ ⊆ (Base‘(ℂfld ↾s (lastS‘𝑣))))
78	76, 77	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℚ ⊆ (Base‘(ℂfld ↾s (lastS‘𝑣))))
79	6	sdrgss 20865	. . . . . . . . . . . . . . . . . . 19 ⊢ ((lastS‘𝑣) ∈ (SubDRing‘ℂfld) → (lastS‘𝑣) ⊆ ℂ)
80	68, 79	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ⊆ ℂ)
81		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
82	81, 6	ressbas2 17288	. . . . . . . . . . . . . . . . . 18 ⊢ ((lastS‘𝑣) ⊆ ℂ → (lastS‘𝑣) = (Base‘(ℂfld ↾s (lastS‘𝑣))))
83	80, 82	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) = (Base‘(ℂfld ↾s (lastS‘𝑣))))
84	78, 83	sseqtrrd 3976	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℚ ⊆ (lastS‘𝑣))
85		simprr 784	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (𝐶‘𝑚) ⊆ (lastS‘𝑣))
86		simpllr 787	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝐴 ∈ (𝐶‘𝑚))
87	85, 86	sseldd 3940	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝐴 ∈ (lastS‘𝑣))
88	87	snssd 4748	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → {𝐴} ⊆ (lastS‘𝑣))
89	84, 88	unssd 4147	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℚ ∪ {𝐴}) ⊆ (lastS‘𝑣))
90	6, 69, 68, 89	fldgenssp 33554	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld fldGen (ℚ ∪ {𝐴})) ⊆ (lastS‘𝑣))
91		id 23	. . . . . . . . . . . . . . . 16 ⊢ ((lastS‘𝑣) ∈ (SubDRing‘ℂfld) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
92	81, 91	subsdrg 33534	. . . . . . . . . . . . . . 15 ⊢ ((lastS‘𝑣) ∈ (SubDRing‘ℂfld) → ((ℂfld fldGen (ℚ ∪ {𝐴})) ∈ (SubDRing‘(ℂfld ↾s (lastS‘𝑣))) ↔ ((ℂfld fldGen (ℚ ∪ {𝐴})) ∈ (SubDRing‘ℂfld) ∧ (ℂfld fldGen (ℚ ∪ {𝐴})) ⊆ (lastS‘𝑣))))
93	92	biimpar 482	. . . . . . . . . . . . . 14 ⊢ (((lastS‘𝑣) ∈ (SubDRing‘ℂfld) ∧ ((ℂfld fldGen (ℚ ∪ {𝐴})) ∈ (SubDRing‘ℂfld) ∧ (ℂfld fldGen (ℚ ∪ {𝐴})) ⊆ (lastS‘𝑣))) → (ℂfld fldGen (ℚ ∪ {𝐴})) ∈ (SubDRing‘(ℂfld ↾s (lastS‘𝑣))))
94	68, 74, 90, 93	syl12anc 849	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld fldGen (ℚ ∪ {𝐴})) ∈ (SubDRing‘(ℂfld ↾s (lastS‘𝑣))))
95	38, 65, 94	sdrgfldext 33957	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (lastS‘𝑣))/FldExt((ℂfld ↾s (lastS‘𝑣)) ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))
96	68	elexd 3480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ∈ V)
97		ressabs 17298	. . . . . . . . . . . . 13 ⊢ (((lastS‘𝑣) ∈ V ∧ (ℂfld fldGen (ℚ ∪ {𝐴})) ⊆ (lastS‘𝑣)) → ((ℂfld ↾s (lastS‘𝑣)) ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))
98	96, 90, 97	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ((ℂfld ↾s (lastS‘𝑣)) ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))
99	95, 98	breqtrd 5131	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))
100	99	ad2antrr 738	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))
101		extdgcl 33963	. . . . . . . . . 10 ⊢ ((ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℕ0*)
102	100, 101	syl 18	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℕ0*)
103		xnn0xr 12573	. . . . . . . . 9 ⊢ (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℕ0* → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℝ*)
104	102, 103	syl 18	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℝ*)
105		extdggt0 33964	. . . . . . . . 9 ⊢ ((ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) → 0 < ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))))
106	100, 105	syl 18	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → 0 < ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))))
107		extdgmul 33970	. . . . . . . . . . 11 ⊢ (((ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) ∧ (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))/FldExt(ℂfld ↾s ℚ)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))))
108	99, 25, 107	syl2anc 595	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))))
109	108	ad2antrr 738	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))))
110		xmulcom 13283	. . . . . . . . . 10 ⊢ ((((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℝ* ∧ ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℝ*) → (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = (((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ·e ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))))
111	104, 37, 110	syl2anc 595	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = (((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ·e ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))))
112	109, 111	eqtrd 2800	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ·e ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))))))
113	35, 37, 104, 106, 112	rexmul2 33011	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℝ)
114		extdggt0 33964	. . . . . . . 8 ⊢ ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))/FldExt(ℂfld ↾s ℚ) → 0 < ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)))
115	26, 114	syl 18	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → 0 < ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)))
116	28, 113, 115	xnn0nnd 33030	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℕ)
117	5, 116	eqeltrid 2869	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (𝐿[:]𝑄) ∈ ℕ)
118	35, 104, 37, 115, 109	rexmul2 33011	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℝ)
119	102, 118	xnn0nn0d 33029	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℕ0)
120	119	nn0zd 12607	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℤ)
121	116	nnnn0d 12556	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℕ0)
122	121	nn0zd 12607	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℤ)
123		rexmul 13288	. . . . . . . . . . 11 ⊢ ((((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℝ ∧ ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℝ) → (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) · ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))))
124	118, 113, 123	syl2anc 595	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ·e ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) · ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))))
125	109, 124	eqtrd 2800	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) · ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))))
126	125	eqcomd 2771	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) · ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)))
127	126, 29	eqtrd 2800	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) · ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = (2↑𝑝))
128		dvds0lem 16314	. . . . . . 7 ⊢ (((((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) ∈ ℤ ∧ ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℤ ∧ (2↑𝑝) ∈ ℤ) ∧ (((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))) · ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∥ (2↑𝑝))
129	120, 122, 33, 127, 128	syl31anc 1396	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∥ (2↑𝑝))
130	5, 129	eqbrtrid 5140	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → (𝐿[:]𝑄) ∥ (2↑𝑝))
131		dvdsprmpweq 16934	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ (𝐿[:]𝑄) ∈ ℕ ∧ 𝑝 ∈ ℕ0) → ((𝐿[:]𝑄) ∥ (2↑𝑝) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛)))
132	131	imp 411	. . . . 5 ⊢ (((2 ∈ ℙ ∧ (𝐿[:]𝑄) ∈ ℕ ∧ 𝑝 ∈ ℕ0) ∧ (𝐿[:]𝑄) ∥ (2↑𝑝)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
133	2, 117, 32, 130, 132	syl31anc 1396	. . . 4 ⊢ (((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑝 ∈ ℕ0) ∧ ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
134	62	simprd 500	. . . 4 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ∃𝑝 ∈ ℕ0 ((ℂfld ↾s (lastS‘𝑣))[:](ℂfld ↾s ℚ)) = (2↑𝑝))
135	133, 134	r19.29a 3173	. . 3 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
136		simplr 780	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝑚 ∈ ω)
137	18, 39, 40, 41, 136	constrextdg2 34056	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)))
138	135, 137	r19.29a 3173	. 2 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
139		constrext2chnlem.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Constr)
140	18	isconstr 34043	. . 3 ⊢ (𝐴 ∈ Constr ↔ ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
141	139, 140	sylib 221	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
142	138, 141	r19.29a 3173	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  {crab 3417  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  ∅c0 4288  {csn 4585  {cpr 4587   class class class wbr 5105  {copab 5167   ↦ cmpt 5186  Oncon0 6350  ‘cfv 6525  (class class class)co 7400  ωcom 7850  reccrdg 8384  ℂcc 11086  ℝcr 11087  0cc0 11088  1c1 11089   + caddc 11091   · cmul 11093  ℝ^*cxr 11230   < clt 11231   − cmin 11429  ℕcn 12224  2c2 12286  ℕ₀cn0 12495  ℕ₀^*cxnn0 12568  ℤcz 12582  ℚcq 12963   ·_e cxmu 13127  ↑cexp 14088  Word cword 14540  lastSclsw 14589  ∗ccj 15137  ℑcim 15139  abscabs 15275   ∥ cdvds 16300  ℙcprime 16719  Basecbs 17259   ↾_s cress 17280   Chain cchn 18651  SubRingcsubrg 20645  DivRingcdr 20804  Fieldcfield 20805  SubDRingcsdrg 20858  ℂ_fldccnfld 21482   fldGen cfldgen 33546  /_FldExtcfldext 33945  [:]cextdg 33947  Constrcconstr 34036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-reg 9542  ax-inf2 9598  ax-ac2 10435  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166  ax-addf 11167  ax-mulf 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-ofr 7665  df-rpss 7710  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-ec 8684  df-qs 8688  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-inf 9391  df-oi 9460  df-r1 9724  df-rank 9725  df-dju 9875  df-card 9913  df-acn 9916  df-ac 10088  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-xnn0 12569  df-z 12583  df-dec 12703  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13128  df-xmul 13130  df-ico 13369  df-fz 13527  df-fzo 13674  df-fl 13816  df-mod 13894  df-seq 14029  df-exp 14089  df-hash 14358  df-word 14541  df-lsw 14590  df-concat 14598  df-s1 14624  df-substr 14669  df-pfx 14699  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-dvds 16301  df-gcd 16543  df-prm 16720  df-pc 16887  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-starv 17315  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ocomp 17321  df-ds 17322  df-unif 17323  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-imas 17552  df-qus 17553  df-mre 17628  df-mrc 17629  df-mri 17630  df-acs 17631  df-proset 18340  df-drs 18341  df-poset 18359  df-ipo 18574  df-chn 18652  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-sbg 18995  df-mulg 19125  df-subg 19180  df-nsg 19181  df-eqg 19182  df-ghm 19275  df-gim 19320  df-cntz 19378  df-oppg 19407  df-lsm 19697  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-srg 20260  df-ring 20308  df-cring 20309  df-oppr 20410  df-dvdsr 20430  df-unit 20431  df-irred 20432  df-invr 20461  df-dvr 20474  df-rhm 20545  df-nzr 20587  df-subrng 20622  df-subrg 20646  df-rlreg 20770  df-domn 20771  df-idom 20772  df-drng 20806  df-field 20807  df-sdrg 20859  df-lmod 20952  df-lss 21022  df-lsp 21062  df-lmhm 21112  df-lmim 21113  df-lmic 21114  df-lbs 21165  df-lvec 21193  df-sra 21263  df-rgmod 21264  df-lidl 21301  df-rsp 21302  df-2idl 21351  df-lpidl 21450  df-lpir 21451  df-pid 21465  df-cnfld 21483  df-dsmm 21842  df-frlm 21857  df-uvc 21893  df-lindf 21916  df-linds 21917  df-assa 21963  df-asp 21964  df-ascl 21965  df-psr 22019  df-mvr 22020  df-mpl 22021  df-opsr 22023  df-evls 22185  df-evl 22186  df-psr1 22300  df-vr1 22301  df-ply1 22302  df-coe1 22303  df-evls1 22436  df-evl1 22437  df-mdeg 26173  df-deg1 26174  df-mon1 26249  df-uc1p 26250  df-q1p 26251  df-r1p 26252  df-ig1p 26253  df-fldgen 33547  df-mxidl 33660  df-dim 33907  df-fldext 33948  df-extdg 33949  df-irng 33991  df-minply 34007  df-constr 34037

This theorem is referenced by:  constrext2chn  34066