Theorem constrext2chnlem 33748

Description: Lemma for constrext2chn 33757. (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 16668	. . . . . 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 7406	. . . . . 6 ⊢ (𝐿[:]𝑄) = ((ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))[:](ℂfld ↾s ℚ))
6		cnfldbas 21274	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
7		eqid 2730	. . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ)
8		eqid 2730	. . . . . . . . . 10 ⊢ (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴}))) = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))
9		cnfldfld 33322	. . . . . . . . . . 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 21316	. . . . . . . . . . . 12 ⊢ ℂfld ∈ DivRing
12		qsubdrg 21342	. . . . . . . . . . . . 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 20703	. . . . . . . . . . . 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 7856	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ω → 𝑚 ∈ On)
20	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ On)
21	18, 20	constrsscn 33738	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐶‘𝑚) ⊆ ℂ)
22	21	sselda 3954	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → 𝐴 ∈ ℂ)
23	22	snssd 4781	. . . . . . . . . . 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 33671	. . . . . . . . 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 33660	. . . . . . . 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 12581	. . . . . . . . . . . 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 14062	. . . . . . . . . 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 2829	. . . . . . . . 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 12654	. . . . . . . 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 12536	. . . . . . . . 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 2730	. . . . . . . . . . . . 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 7410	. . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain(SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) = (∅‘0))
48		0fv 6909	. . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . 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 12208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
53		nnq 12935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℚ
55	54	ne0ii 4315	. . . . . . . . . . . . . . . . . . . . . 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 2994	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain(SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) ∧ 𝑣 = ∅) → (𝑣‘0) ≠ ∅)
58	57	neneqd 2932	. . . . . . . . . . . . . . . . . . 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 2934	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain(SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ≠ ∅)
61	42, 60	hashne0 32743	. . . . . . . . . . . . . . . 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 33726	. . . . . . . . . . . . . . 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 33651	. . . . . . . . . . . . . 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 32941	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain(SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝑣 ∈ Word (SubDRing‘ℂfld))
67		lswcl 14543	. . . . . . . . . . . . . . 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 12933	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ⊆ ℂ
71	70	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ℚ ⊆ ℂ)
72	71, 23	unssd 4163	. . . . . . . . . . . . . . . 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 33272	. . . . . . . . . . . . . 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 27537	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ = (Base‘(ℂfld ↾s ℚ))
76	75, 63	fldextsdrg 33658	. . . . . . . . . . . . . . . . . 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 20708	. . . . . . . . . . . . . . . . . 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 20708	. . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
82	81, 6	ressbas2 17214	. . . . . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain(SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → 𝐴 ∈ (lastS‘𝑣))
88	87	snssd 4781	. . . . . . . . . . . . . . . 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 4163	. . . . . . . . . . . . . . 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 33276	. . . . . . . . . . . . . 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 33256	. . . . . . . . . . . . . . 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 33654	. . . . . . . . . . . 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 3479	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) ∧ 𝑣 ∈ ( < Chain(SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣))) → (lastS‘𝑣) ∈ V)
97		ressabs 17224	. . . . . . . . . . . . 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 5141	. . . . . . . . . . 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 33660	. . . . . . . . . 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 12536	. . . . . . . . 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 33661	. . . . . . . . 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 33667	. . . . . . . . . . 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 13239	. . . . . . . . . 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 2765	. . . . . . . 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 32685	. . . . . . 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 33661	. . . . . . . 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 32704	. . . . . 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 2833	. . . . 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 32685	. . . . . . . . 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 32703	. . . . . . . 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 12571	. . . . . . 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 12519	. . . . . . . 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 12571	. . . . . . 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 13244	. . . . . . . . . . 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 2765	. . . . . . . . 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 2736	. . . . . . . 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 2765	. . . . . . 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 16243	. . . . . . 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 5150	. . . . 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 16861	. . . . . 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 3143	. . 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 33747	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑣 ∈ ( < Chain(SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑚) ⊆ (lastS‘𝑣)))
138	135, 137	r19.29a 3143	. 2 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝐴 ∈ (𝐶‘𝑚)) → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
139		constrext2chnlem.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Constr)
140	18	isconstr 33734	. . 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 3143	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2927  ∃wrex 3055  {crab 3411  Vcvv 3455   ∪ cun 3920   ⊆ wss 3922  ∅c0 4304  {csn 4597  {cpr 4599   class class class wbr 5115  {copab 5177   ↦ cmpt 5196  Oncon0 6340  ‘cfv 6519  (class class class)co 7394  ωcom 7850  reccrdg 8386  ℂcc 11084  ℝcr 11085  0cc0 11086  1c1 11087   + caddc 11089   · cmul 11091  ℝ^*cxr 11225   < clt 11226   − cmin 11423  ℕcn 12197  2c2 12252  ℕ₀cn0 12458  ℕ₀^*cxnn0 12531  ℤcz 12545  ℚcq 12921   ·_e cxmu 13084  ↑cexp 14036  Word cword 14488  lastSclsw 14537  ∗ccj 15072  ℑcim 15074  abscabs 15210   ∥ cdvds 16229  ℙcprime 16647  Basecbs 17185   ↾_s cress 17206  SubRingcsubrg 20484  DivRingcdr 20644  Fieldcfield 20645  SubDRingcsdrg 20701  ℂ_fldccnfld 21270  Chaincchn 32938   fldGen cfldgen 33268  /_FldExtcfldext 33642  [:]cextdg 33644  Constrcconstr 33727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718  ax-reg 9563  ax-inf2 9612  ax-ac2 10434  ax-cnex 11142  ax-resscn 11143  ax-1cn 11144  ax-icn 11145  ax-addcl 11146  ax-addrcl 11147  ax-mulcl 11148  ax-mulrcl 11149  ax-mulcom 11150  ax-addass 11151  ax-mulass 11152  ax-distr 11153  ax-i2m1 11154  ax-1ne0 11155  ax-1rid 11156  ax-rnegex 11157  ax-rrecex 11158  ax-cnre 11159  ax-pre-lttri 11160  ax-pre-lttrn 11161  ax-pre-ltadd 11162  ax-pre-mulgt0 11163  ax-pre-sup 11164  ax-addf 11165  ax-mulf 11166

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-nel 3032  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-tp 4602  df-op 4604  df-uni 4880  df-int 4919  df-iun 4965  df-iin 4966  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-se 5600  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-isom 6528  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7660  df-ofr 7661  df-rpss 7706  df-om 7851  df-1st 7977  df-2nd 7978  df-supp 8149  df-tpos 8214  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-1o 8443  df-2o 8444  df-oadd 8447  df-er 8682  df-ec 8684  df-qs 8688  df-map 8805  df-pm 8806  df-ixp 8875  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-fsupp 9331  df-sup 9411  df-inf 9412  df-oi 9481  df-r1 9735  df-rank 9736  df-dju 9872  df-card 9910  df-acn 9913  df-ac 10087  df-pnf 11228  df-mnf 11229  df-xr 11230  df-ltxr 11231  df-le 11232  df-sub 11425  df-neg 11426  df-div 11852  df-nn 12198  df-2 12260  df-3 12261  df-4 12262  df-5 12263  df-6 12264  df-7 12265  df-8 12266  df-9 12267  df-n0 12459  df-xnn0 12532  df-z 12546  df-dec 12666  df-uz 12810  df-q 12922  df-rp 12966  df-xneg 13085  df-xmul 13087  df-ico 13325  df-fz 13482  df-fzo 13629  df-fl 13766  df-mod 13844  df-seq 13977  df-exp 14037  df-hash 14306  df-word 14489  df-lsw 14538  df-concat 14546  df-s1 14571  df-substr 14616  df-pfx 14646  df-cj 15075  df-re 15076  df-im 15077  df-sqrt 15211  df-abs 15212  df-dvds 16230  df-gcd 16471  df-prm 16648  df-pc 16814  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-ress 17207  df-plusg 17239  df-mulr 17240  df-starv 17241  df-sca 17242  df-vsca 17243  df-ip 17244  df-tset 17245  df-ple 17246  df-ocomp 17247  df-ds 17248  df-unif 17249  df-hom 17250  df-cco 17251  df-0g 17410  df-gsum 17411  df-prds 17416  df-pws 17418  df-imas 17477  df-qus 17478  df-mre 17553  df-mrc 17554  df-mri 17555  df-acs 17556  df-proset 18261  df-drs 18262  df-poset 18280  df-ipo 18493  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18716  df-submnd 18717  df-grp 18874  df-minusg 18875  df-sbg 18876  df-mulg 19006  df-subg 19061  df-nsg 19062  df-eqg 19063  df-ghm 19151  df-gim 19197  df-cntz 19255  df-oppg 19284  df-lsm 19572  df-cmn 19718  df-abl 19719  df-mgp 20056  df-rng 20068  df-ur 20097  df-srg 20102  df-ring 20150  df-cring 20151  df-oppr 20252  df-dvdsr 20272  df-unit 20273  df-irred 20274  df-invr 20303  df-dvr 20316  df-rhm 20387  df-nzr 20428  df-subrng 20461  df-subrg 20485  df-rlreg 20609  df-domn 20610  df-idom 20611  df-drng 20646  df-field 20647  df-sdrg 20702  df-lmod 20774  df-lss 20844  df-lsp 20884  df-lmhm 20935  df-lmim 20936  df-lmic 20937  df-lbs 20988  df-lvec 21016  df-sra 21086  df-rgmod 21087  df-lidl 21124  df-rsp 21125  df-2idl 21166  df-lpidl 21238  df-lpir 21239  df-pid 21253  df-cnfld 21271  df-dsmm 21647  df-frlm 21662  df-uvc 21698  df-lindf 21721  df-linds 21722  df-assa 21768  df-asp 21769  df-ascl 21770  df-psr 21824  df-mvr 21825  df-mpl 21826  df-opsr 21828  df-evls 21987  df-evl 21988  df-psr1 22070  df-vr1 22071  df-ply1 22072  df-coe1 22073  df-evls1 22208  df-evl1 22209  df-mdeg 25967  df-deg1 25968  df-mon1 26043  df-uc1p 26044  df-q1p 26045  df-r1p 26046  df-ig1p 26047  df-chn 32939  df-fldgen 33269  df-mxidl 33439  df-dim 33603  df-fldext 33645  df-extdg 33646  df-irng 33687  df-minply 33698  df-constr 33728

This theorem is referenced by:  constrext2chn  33757