Theorem constrext2chnlem 33758

Description: Lemma for constrext2chn 33767. (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 16600	. . . . . 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 7358	. . . . . 6 ⊢ (𝐿[:]𝑄) = ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))
6		cnfldbas 21293	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
7		eqid 2731	. . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ)
8		eqid 2731	. . . . . . . . . 10 ⊢ (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
9		cnfldfld 33302	. . . . . . . . . . 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 21333	. . . . . . . . . . . 12 ⊢ ℂfld ∈ DivRing
12		qsubdrg 21354	. . . . . . . . . . . . 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 20701	. . . . . . . . . . . 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 7802	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ω → 𝑚 ∈ On)
20	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ On)
21	18, 20	constrsscn 33748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐶‘𝑚) ⊆ ℂ)
22	21	sselda 3934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝐴 ∈ ℂ)
23	22	snssd 4761	. . . . . . . . . . 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 33676	. . . . . . . . 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 33664	. . . . . . . 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 12501	. . . . . . . . . . . 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 13991	. . . . . . . . . 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 2831	. . . . . . . . 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 12574	. . . . . . . 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 12456	. . . . . . . . 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 2731	. . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = (∅‘0))
48		0fv 6863	. . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . 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 12133	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
53		nnq 12857	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℚ
55	54	ne0ii 4294	. . . . . . . . . . . . . . . . . . . . . 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 2995	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) ≠ ∅)
58	57	neneqd 2933	. . . . . . . . . . . . . . . . . . 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 2935	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ≠ ∅)
61	42, 60	hashne0 32787	. . . . . . . . . . . . . . . 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 33736	. . . . . . . . . . . . . . 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 33655	. . . . . . . . . . . . . 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 18511	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ Word (SubDRing‘ℂfld))
67		lswcl 14472	. . . . . . . . . . . . . . 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 12855	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ⊆ ℂ
71	70	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ℚ ⊆ ℂ)
72	71, 23	unssd 4142	. . . . . . . . . . . . . . . 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 33275	. . . . . . . . . . . . . 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 27555	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ = (Base‘(ℂfld ↾s ℚ))
76	75, 63	fldextsdrg 33662	. . . . . . . . . . . . . . . . . 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 20706	. . . . . . . . . . . . . . . . . 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 20706	. . . . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
82	81, 6	ressbas2 17146	. . . . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝐴 ∈ (lastS‘𝑣))
88	87	snssd 4761	. . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . 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 33279	. . . . . . . . . . . . . 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 33259	. . . . . . . . . . . . . . 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 33658	. . . . . . . . . . . 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 3460	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ∈ V)
97		ressabs 17156	. . . . . . . . . . . . 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 5117	. . . . . . . . . . 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 33664	. . . . . . . . . 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 12456	. . . . . . . . 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 33665	. . . . . . . . 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 33671	. . . . . . . . . . 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 13162	. . . . . . . . . 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 2766	. . . . . . . 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 32732	. . . . . . 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 33665	. . . . . . . 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 32751	. . . . . 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 2835	. . . . 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 32732	. . . . . . . . 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 32750	. . . . . . . 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 12491	. . . . . . 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 12439	. . . . . . . 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 12491	. . . . . . 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 13167	. . . . . . . . . . 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 2766	. . . . . . . . 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 2737	. . . . . . . 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 2766	. . . . . . 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 16174	. . . . . . 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 5126	. . . . 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 16793	. . . . . 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 3140	. . 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 33757	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)))
138	135, 137	r19.29a 3140	. 2 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
139		constrext2chnlem.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Constr)
140	18	isconstr 33744	. . 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 3140	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  {crab 3395  Vcvv 3436   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  {csn 4576  {cpr 4578   class class class wbr 5091  {copab 5153   ↦ cmpt 5172  Oncon0 6306  ‘cfv 6481  (class class class)co 7346  ωcom 7796  reccrdg 8328  ℂcc 11001  ℝcr 11002  0cc0 11003  1c1 11004   + caddc 11006   · cmul 11008  ℝ^*cxr 11142   < clt 11143   − cmin 11341  ℕcn 12122  2c2 12177  ℕ₀cn0 12378  ℕ₀^*cxnn0 12451  ℤcz 12465  ℚcq 12843   ·_e cxmu 13007  ↑cexp 13965  Word cword 14417  lastSclsw 14466  ∗ccj 15000  ℑcim 15002  abscabs 15138   ∥ cdvds 16160  ℙcprime 16579  Basecbs 17117   ↾_s cress 17138   Chain cchn 18508  SubRingcsubrg 20482  DivRingcdr 20642  Fieldcfield 20643  SubDRingcsdrg 20699  ℂ_fldccnfld 21289   fldGen cfldgen 33271  /_FldExtcfldext 33646  [:]cextdg 33648  Constrcconstr 33737

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-reg 9478  ax-inf2 9531  ax-ac2 10351  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081  ax-addf 11082  ax-mulf 11083

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-ofr 7611  df-rpss 7656  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-ec 8624  df-qs 8628  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-sup 9326  df-inf 9327  df-oi 9396  df-r1 9654  df-rank 9655  df-dju 9791  df-card 9829  df-acn 9832  df-ac 10004  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-xnn0 12452  df-z 12466  df-dec 12586  df-uz 12730  df-q 12844  df-rp 12888  df-xneg 13008  df-xmul 13010  df-ico 13248  df-fz 13405  df-fzo 13552  df-fl 13693  df-mod 13771  df-seq 13906  df-exp 13966  df-hash 14235  df-word 14418  df-lsw 14467  df-concat 14475  df-s1 14501  df-substr 14546  df-pfx 14576  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-dvds 16161  df-gcd 16403  df-prm 16580  df-pc 16746  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-mulr 17172  df-starv 17173  df-sca 17174  df-vsca 17175  df-ip 17176  df-tset 17177  df-ple 17178  df-ocomp 17179  df-ds 17180  df-unif 17181  df-hom 17182  df-cco 17183  df-0g 17342  df-gsum 17343  df-prds 17348  df-pws 17350  df-imas 17409  df-qus 17410  df-mre 17485  df-mrc 17486  df-mri 17487  df-acs 17488  df-proset 18197  df-drs 18198  df-poset 18216  df-ipo 18431  df-chn 18509  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-mhm 18688  df-submnd 18689  df-grp 18846  df-minusg 18847  df-sbg 18848  df-mulg 18978  df-subg 19033  df-nsg 19034  df-eqg 19035  df-ghm 19123  df-gim 19169  df-cntz 19227  df-oppg 19256  df-lsm 19546  df-cmn 19692  df-abl 19693  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-cring 20152  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-irred 20275  df-invr 20304  df-dvr 20317  df-rhm 20388  df-nzr 20426  df-subrng 20459  df-subrg 20483  df-rlreg 20607  df-domn 20608  df-idom 20609  df-drng 20644  df-field 20645  df-sdrg 20700  df-lmod 20793  df-lss 20863  df-lsp 20903  df-lmhm 20954  df-lmim 20955  df-lmic 20956  df-lbs 21007  df-lvec 21035  df-sra 21105  df-rgmod 21106  df-lidl 21143  df-rsp 21144  df-2idl 21185  df-lpidl 21257  df-lpir 21258  df-pid 21272  df-cnfld 21290  df-dsmm 21667  df-frlm 21682  df-uvc 21718  df-lindf 21741  df-linds 21742  df-assa 21788  df-asp 21789  df-ascl 21790  df-psr 21844  df-mvr 21845  df-mpl 21846  df-opsr 21848  df-evls 22007  df-evl 22008  df-psr1 22090  df-vr1 22091  df-ply1 22092  df-coe1 22093  df-evls1 22228  df-evl1 22229  df-mdeg 25985  df-deg1 25986  df-mon1 26061  df-uc1p 26062  df-q1p 26063  df-r1p 26064  df-ig1p 26065  df-fldgen 33272  df-mxidl 33420  df-dim 33607  df-fldext 33649  df-extdg 33650  df-irng 33692  df-minply 33708  df-constr 33738

This theorem is referenced by:  constrext2chn  33767