Theorem constrext2chnlem 33907

Description: Lemma for constrext2chn 33916. (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 16619	. . . . . 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 7370	. . . . . 6 ⊢ (𝐿[:]𝑄) = ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))
6		cnfldbas 21313	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
7		eqid 2736	. . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ)
8		eqid 2736	. . . . . . . . . 10 ⊢ (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
9		cnfldfld 33423	. . . . . . . . . . 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 21353	. . . . . . . . . . . 12 ⊢ ℂfld ∈ DivRing
12		qsubdrg 21374	. . . . . . . . . . . . 13 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
13	12	simpli 483	. . . . . . . . . . . 12 ⊢ ℚ ∈ (SubRing‘ℂfld)
14	12	simpri 485	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
15		issdrg 20721	. . . . . . . . . . . 12 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
16	11, 13, 14, 15	mpbir3an 1342	. . . . . . . . . . 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 7814	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ω → 𝑚 ∈ On)
20	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ On)
21	18, 20	constrsscn 33897	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐶‘𝑚) ⊆ ℂ)
22	21	sselda 3933	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝐴 ∈ ℂ)
23	22	snssd 4765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → {𝐴} ⊆ ℂ)
24	23	ad2antrr 726	. . . . . . . . . 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 33825	. . . . . . . . 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 726	. . . . . . . 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 33813	. . . . . . . 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 484	. . . . . . . . . 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 12523	. . . . . . . . . . . 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 768	. . . . . . . . . . 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 14010	. . . . . . . . . 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 2836	. . . . . . . . 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 12596	. . . . . . . 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 12479	. . . . . . . . 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 2736	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ ( < Chain (SubDRing‘ℂfld)))
43		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (𝑣‘0) = ℚ)
44	43	oveq2d 7374	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → 𝑣 = ∅)
47	46	fveq1d 6836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = (∅‘0))
48		0fv 6875	. . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = ∅)
51	43	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = ℚ)
52		1nn 12156	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
53		nnq 12875	. . . . . . . . . . . . . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) ≠ ∅)
58	57	neneqd 2937	. . . . . . . . . . . . . . . . . . 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 816	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ¬ 𝑣 = ∅)
60	59	neqned 2939	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ≠ ∅)
61	42, 60	hashne0 32890	. . . . . . . . . . . . . . . 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 33885	. . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . 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 33804	. . . . . . . . . . . . . 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 18531	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ Word (SubDRing‘ℂfld))
67		lswcl 14491	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ 𝑣 ≠ ∅) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
68	66, 60, 67	syl2anc 584	. . . . . . . . . . . . . 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 12873	. . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 33396	. . . . . . . . . . . . . 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 27586	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ = (Base‘(ℂfld ↾s ℚ))
76	75, 63	fldextsdrg 33811	. . . . . . . . . . . . . . . . . 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 20726	. . . . . . . . . . . . . . . . . 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 20726	. . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
82	81, 6	ressbas2 17165	. . . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → ℚ ⊆ (lastS‘𝑣))
85		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (𝐶‘𝑚) ⊆ (lastS‘𝑣))
86		simpllr 775	. . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝐴 ∈ (lastS‘𝑣))
88	87	snssd 4765	. . . . . . . . . . . . . . . 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 33400	. . . . . . . . . . . . . 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 33380	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 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 33807	. . . . . . . . . . . 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 3464	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ∈ V)
97		ressabs 17175	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 5124	. . . . . . . . . . 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 726	. . . . . . . . . 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 33813	. . . . . . . . . 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 12479	. . . . . . . . 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 33814	. . . . . . . . 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 33820	. . . . . . . . . . 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 584	. . . . . . . . . 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 726	. . . . . . . . 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 13181	. . . . . . . . . 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 584	. . . . . . . . 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 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 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 32834	. . . . . . 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 33814	. . . . . . . 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 32853	. . . . . 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 2840	. . . . 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 32834	. . . . . . . . 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 32852	. . . . . . . 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 12513	. . . . . . 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 12462	. . . . . . . 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 12513	. . . . . . 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 13186	. . . . . . . . . . 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 584	. . . . . . . . . 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 2771	. . . . . . . . 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 2742	. . . . . . . 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 2771	. . . . . . 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 16193	. . . . . . 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 1375	. . . . . 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 5133	. . . . 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 16812	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ (𝐿[:]𝑄) ∈ ℕ ∧ 𝑝 ∈ ℕ0) → ((𝐿[:]𝑄) ∥ (2↑𝑝) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛)))
132	131	imp 406	. . . . 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 1375	. . . 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 495	. . . 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 3144	. . 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 768	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝑚 ∈ ω)
137	18, 39, 40, 41, 136	constrextdg2 33906	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)))
138	135, 137	r19.29a 3144	. 2 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
139		constrext2chnlem.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Constr)
140	18	isconstr 33893	. . 3 ⊢ (𝐴 ∈ Constr ↔ ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
141	139, 140	sylib 218	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
142	138, 141	r19.29a 3144	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060  {crab 3399  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580  {cpr 4582   class class class wbr 5098  {copab 5160   ↦ cmpt 5179  Oncon0 6317  ‘cfv 6492  (class class class)co 7358  ωcom 7808  reccrdg 8340  ℂcc 11024  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029   · cmul 11031  ℝ^*cxr 11165   < clt 11166   − cmin 11364  ℕcn 12145  2c2 12200  ℕ₀cn0 12401  ℕ₀^*cxnn0 12474  ℤcz 12488  ℚcq 12861   ·_e cxmu 13025  ↑cexp 13984  Word cword 14436  lastSclsw 14485  ∗ccj 15019  ℑcim 15021  abscabs 15157   ∥ cdvds 16179  ℙcprime 16598  Basecbs 17136   ↾_s cress 17157   Chain cchn 18528  SubRingcsubrg 20502  DivRingcdr 20662  Fieldcfield 20663  SubDRingcsdrg 20719  ℂ_fldccnfld 21309   fldGen cfldgen 33392  /_FldExtcfldext 33795  [:]cextdg 33797  Constrcconstr 33886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550  ax-ac2 10373  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105  ax-mulf 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-rpss 7668  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-ec 8637  df-qs 8641  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-inf 9346  df-oi 9415  df-r1 9676  df-rank 9677  df-dju 9813  df-card 9851  df-acn 9854  df-ac 10026  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xmul 13028  df-ico 13267  df-fz 13424  df-fzo 13571  df-fl 13712  df-mod 13790  df-seq 13925  df-exp 13985  df-hash 14254  df-word 14437  df-lsw 14486  df-concat 14494  df-s1 14520  df-substr 14565  df-pfx 14595  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-dvds 16180  df-gcd 16422  df-prm 16599  df-pc 16765  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ocomp 17198  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-imas 17429  df-qus 17430  df-mre 17505  df-mrc 17506  df-mri 17507  df-acs 17508  df-proset 18217  df-drs 18218  df-poset 18236  df-ipo 18451  df-chn 18529  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-nsg 19054  df-eqg 19055  df-ghm 19142  df-gim 19188  df-cntz 19246  df-oppg 19275  df-lsm 19565  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-irred 20295  df-invr 20324  df-dvr 20337  df-rhm 20408  df-nzr 20446  df-subrng 20479  df-subrg 20503  df-rlreg 20627  df-domn 20628  df-idom 20629  df-drng 20664  df-field 20665  df-sdrg 20720  df-lmod 20813  df-lss 20883  df-lsp 20923  df-lmhm 20974  df-lmim 20975  df-lmic 20976  df-lbs 21027  df-lvec 21055  df-sra 21125  df-rgmod 21126  df-lidl 21163  df-rsp 21164  df-2idl 21205  df-lpidl 21277  df-lpir 21278  df-pid 21292  df-cnfld 21310  df-dsmm 21687  df-frlm 21702  df-uvc 21738  df-lindf 21761  df-linds 21762  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evls1 22259  df-evl1 22260  df-mdeg 26016  df-deg1 26017  df-mon1 26092  df-uc1p 26093  df-q1p 26094  df-r1p 26095  df-ig1p 26096  df-fldgen 33393  df-mxidl 33541  df-dim 33756  df-fldext 33798  df-extdg 33799  df-irng 33841  df-minply 33857  df-constr 33887

This theorem is referenced by:  constrext2chn  33916