Theorem constrext2chnlem 34008

Description: Lemma for constrext2chn 34017. (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 16709	. . . . . 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 7404	. . . . . 6 ⊢ (𝐿[:]𝑄) = ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))
6		cnfldbas 21408	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
7		eqid 2761	. . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ)
8		eqid 2761	. . . . . . . . . 10 ⊢ (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
9		cnfldfld 33489	. . . . . . . . . . 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 21433	. . . . . . . . . . . 12 ⊢ ℂfld ∈ DivRing
12		qsubdrg 21451	. . . . . . . . . . . . 13 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
13	12	simpli 487	. . . . . . . . . . . 12 ⊢ ℚ ∈ (SubRing‘ℂfld)
14	12	simpri 489	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
15		issdrg 20817	. . . . . . . . . . . 12 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
16	11, 13, 14, 15	mpbir3an 1354	. . . . . . . . . . 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 7848	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ω → 𝑚 ∈ On)
20	19	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ On)
21	18, 20	constrsscn 33998	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐶‘𝑚) ⊆ ℂ)
22	21	sselda 3936	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝐴 ∈ ℂ)
23	22	snssd 4744	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → {𝐴} ⊆ ℂ)
24	23	ad2antrr 736	. . . . . . . . . 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 33926	. . . . . . . . 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 736	. . . . . . . 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 33914	. . . . . . . 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 17	. . . . . . 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 488	. . . . . . . . . 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 12600	. . . . . . . . . . . 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 778	. . . . . . . . . . 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 14097	. . . . . . . . . 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 2861	. . . . . . . . 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 12674	. . . . . . . 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 12556	. . . . . . . . 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 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 (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ)) ∈ ℝ*)
38		eqid 2761	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ ( < Chain (SubDRing‘ℂfld)))
43		simprl 780	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (𝑣‘0) = ℚ)
44	43	oveq2d 7408	. . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → 𝑣 = ∅)
47	46	fveq1d 6865	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = (∅‘0))
48		0fv 6904	. . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = ∅)
51	43	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = ℚ)
52		1nn 12218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
53		nnq 12960	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℚ
55	54	ne0ii 4296	. . . . . . . . . . . . . . . . . . . . . 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 3023	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) ≠ ∅)
58	57	neneqd 2961	. . . . . . . . . . . . . . . . . . 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 826	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ¬ 𝑣 = ∅)
60	59	neqned 2963	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ≠ ∅)
61	42, 60	hashne0 32962	. . . . . . . . . . . . . . . 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 33986	. . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . 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 33905	. . . . . . . . . . . . . 14 ⊢ ((ℂfld ↾s (lastS‘𝑣))/FldExt(ℂfld ↾s ℚ) → (ℂfld ↾s (lastS‘𝑣)) ∈ Field)
65	63, 64	syl 17	. . . . . . . . . . . . 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 18623	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ Word (SubDRing‘ℂfld))
67		lswcl 14578	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ 𝑣 ≠ ∅) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
68	66, 60, 67	syl2anc 593	. . . . . . . . . . . . . 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 12958	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ⊆ ℂ
71	70	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ℚ ⊆ ℂ)
72	71, 23	unssd 4144	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → (ℚ ∪ {𝐴}) ⊆ ℂ)
73	72	ad2antrr 736	. . . . . . . . . . . . . . 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 33462	. . . . . . . . . . . . . 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 27660	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ = (Base‘(ℂfld ↾s ℚ))
76	75, 63	fldextsdrg 33912	. . . . . . . . . . . . . . . . . 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 20822	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubDRing‘(ℂfld ↾s (lastS‘𝑣))) → ℚ ⊆ (Base‘(ℂfld ↾s (lastS‘𝑣))))
78	76, 77	syl 17	. . . . . . . . . . . . . . . . 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 20822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((lastS‘𝑣) ∈ (SubDRing‘ℂfld) → (lastS‘𝑣) ⊆ ℂ)
80	68, 79	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ⊆ ℂ)
81		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
82	81, 6	ressbas2 17257	. . . . . . . . . . . . . . . . . 18 ⊢ ((lastS‘𝑣) ⊆ ℂ → (lastS‘𝑣) = (Base‘(ℂfld ↾s (lastS‘𝑣))))
83	80, 82	syl 17	. . . . . . . . . . . . . . . . 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 3973	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℚ ⊆ (lastS‘𝑣))
85		simprr 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (𝐶‘𝑚) ⊆ (lastS‘𝑣))
86		simpllr 785	. . . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝐴 ∈ (lastS‘𝑣))
88	87	snssd 4744	. . . . . . . . . . . . . . . 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 4144	. . . . . . . . . . . . . . 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 33466	. . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . 16 ⊢ ((lastS‘𝑣) ∈ (SubDRing‘ℂfld) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
92	81, 91	subsdrg 33446	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 847	. . . . . . . . . . . . 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 33908	. . . . . . . . . . . 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 3476	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ∈ V)
97		ressabs 17267	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 5125	. . . . . . . . . . 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 736	. . . . . . . . . 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 33914	. . . . . . . . . 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 17	. . . . . . . . 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 12556	. . . . . . . . 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 17	. . . . . . . 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 33915	. . . . . . . . 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 17	. . . . . . . 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 33921	. . . . . . . . . . 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 593	. . . . . . . . . 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 736	. . . . . . . . 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 13266	. . . . . . . . . 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 593	. . . . . . . . 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 2796	. . . . . . . 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 32906	. . . . . . 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 33915	. . . . . . . 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 17	. . . . . . 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 32925	. . . . . 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 2865	. . . . 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 32906	. . . . . . . . 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 32924	. . . . . . . 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 12590	. . . . . . 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 12539	. . . . . . . 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 12590	. . . . . . 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 13271	. . . . . . . . . . 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 593	. . . . . . . . . 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 2796	. . . . . . . . 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 2767	. . . . . . . 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 2796	. . . . . . 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 16283	. . . . . . 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 1391	. . . . . 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 5134	. . . . 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 16903	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ (𝐿[:]𝑄) ∈ ℕ ∧ 𝑝 ∈ ℕ0) → ((𝐿[:]𝑄) ∥ (2↑𝑝) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛)))
132	131	imp 410	. . . . 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 1391	. . . 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 499	. . . 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 3169	. . 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 778	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝑚 ∈ ω)
137	18, 39, 40, 41, 136	constrextdg2 34007	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)))
138	135, 137	r19.29a 3169	. 2 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
139		constrext2chnlem.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Constr)
140	18	isconstr 33994	. . 3 ⊢ (𝐴 ∈ Constr ↔ ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
141	139, 140	sylib 220	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
142	138, 141	r19.29a 3169	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))

Syntax hints:   → wi 4   ∧ wa 399   ∨ w3o 1096   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413  Vcvv 3453   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4581  {cpr 4583   class class class wbr 5099  {copab 5161   ↦ cmpt 5180  Oncon0 6342  ‘cfv 6517  (class class class)co 7392  ωcom 7842  reccrdg 8375  ℂcc 11068  ℝcr 11069  0cc0 11070  1c1 11071   + caddc 11073   · cmul 11075  ℝ^*cxr 11212   < clt 11213   − cmin 11411  ℕcn 12207  2c2 12269  ℕ₀cn0 12478  ℕ₀^*cxnn0 12551  ℤcz 12565  ℚcq 12946   ·_e cxmu 13110  ↑cexp 14071  Word cword 14523  lastSclsw 14572  ∗ccj 15106  ℑcim 15108  abscabs 15244   ∥ cdvds 16269  ℙcprime 16688  Basecbs 17228   ↾_s cress 17249   Chain cchn 18620  SubRingcsubrg 20598  DivRingcdr 20758  Fieldcfield 20759  SubDRingcsdrg 20815  ℂ_fldccnfld 21404   fldGen cfldgen 33458  /_FldExtcfldext 33896  [:]cextdg 33898  Constrcconstr 33987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-reg 9537  ax-inf2 9593  ax-ac2 10417  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148  ax-addf 11149  ax-mulf 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-ofr 7657  df-rpss 7702  df-om 7843  df-1st 7966  df-2nd 7967  df-supp 8136  df-tpos 8201  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-er 8673  df-ec 8675  df-qs 8679  df-map 8805  df-pm 8806  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fsupp 9305  df-sup 9385  df-inf 9386  df-oi 9455  df-r1 9719  df-rank 9720  df-dju 9856  df-card 9894  df-acn 9897  df-ac 10069  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-xnn0 12552  df-z 12566  df-dec 12686  df-uz 12837  df-q 12947  df-rp 12991  df-xneg 13111  df-xmul 13113  df-ico 13352  df-fz 13510  df-fzo 13657  df-fl 13799  df-mod 13877  df-seq 14012  df-exp 14072  df-hash 14341  df-word 14524  df-lsw 14573  df-concat 14581  df-s1 14607  df-substr 14652  df-pfx 14682  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-dvds 16270  df-gcd 16512  df-prm 16689  df-pc 16856  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-starv 17284  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ocomp 17290  df-ds 17291  df-unif 17292  df-hom 17293  df-cco 17294  df-0g 17453  df-gsum 17454  df-prds 17459  df-pws 17461  df-imas 17521  df-qus 17522  df-mre 17597  df-mrc 17598  df-mri 17599  df-acs 17600  df-proset 18309  df-drs 18310  df-poset 18328  df-ipo 18543  df-chn 18621  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800  df-submnd 18801  df-grp 18961  df-minusg 18962  df-sbg 18963  df-mulg 19093  df-subg 19148  df-nsg 19149  df-eqg 19150  df-ghm 19237  df-gim 19282  df-cntz 19340  df-oppg 19369  df-lsm 19659  df-cmn 19805  df-abl 19806  df-mgp 20170  df-rng 20182  df-ur 20211  df-srg 20216  df-ring 20264  df-cring 20265  df-oppr 20365  df-dvdsr 20385  df-unit 20386  df-irred 20387  df-invr 20416  df-dvr 20429  df-rhm 20500  df-nzr 20542  df-subrng 20575  df-subrg 20599  df-rlreg 20723  df-domn 20724  df-idom 20725  df-drng 20760  df-field 20761  df-sdrg 20816  df-lmod 20909  df-lss 20979  df-lsp 21019  df-lmhm 21069  df-lmim 21070  df-lmic 21071  df-lbs 21122  df-lvec 21150  df-sra 21220  df-rgmod 21221  df-lidl 21258  df-rsp 21259  df-2idl 21300  df-lpidl 21372  df-lpir 21373  df-pid 21387  df-cnfld 21405  df-dsmm 21764  df-frlm 21779  df-uvc 21815  df-lindf 21838  df-linds 21839  df-assa 21885  df-asp 21886  df-ascl 21887  df-psr 21941  df-mvr 21942  df-mpl 21943  df-opsr 21945  df-evls 22107  df-evl 22108  df-psr1 22222  df-vr1 22223  df-ply1 22224  df-coe1 22225  df-evls1 22358  df-evl1 22359  df-mdeg 26095  df-deg1 26096  df-mon1 26171  df-uc1p 26172  df-q1p 26173  df-r1p 26174  df-ig1p 26175  df-fldgen 33459  df-mxidl 33609  df-dim 33858  df-fldext 33899  df-extdg 33900  df-irng 33942  df-minply 33958  df-constr 33988

This theorem is referenced by:  constrext2chn  34017