fldext2chn - Mathbox for Thierry Arnoux

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Theorem fldext2chn 34430

Description: In a non-empty chain 𝑇 of quadratic field extensions, the degree of the final extension is always a power of two. (Contributed by Thierry Arnoux, 19-Oct-2025.)

**Hypotheses**
Ref	Expression
fldext2chn.e	⊢ 𝐸 = (𝑊 ↾_s 𝑒)
fldext2chn.f	⊢ 𝐹 = (𝑊 ↾_s 𝑓)
fldext2chn.l	⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
fldext2chn.t	⊢ (𝜑 → 𝑇 ∈ ( < Chain (SubDRing‘𝑊)))
fldext2chn.w	⊢ (𝜑 → 𝑊 ∈ Field)
fldext2chn.1	⊢ (𝜑 → (𝑊 ↾_s (𝑇‘0)) = 𝑄)
fldext2chn.2	⊢ (𝜑 → (𝑊 ↾_s (lastS‘𝑇)) = 𝐿)
fldext2chn.3	⊢ (𝜑 → 0 < (♯‘𝑇))

**Assertion**
Ref	Expression
fldext2chn	⊢ (𝜑 → (𝐿/_FldExt𝑄 ∧ ∃𝑛 ∈ ℕ₀ (𝐿[:]𝑄) = (2↑𝑛)))

Distinct variable groups: 𝑇,𝑛 𝑛,𝑊 𝑒,𝑊,𝑓 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑒, 𝑓) 𝑄(𝑒, 𝑓, 𝑛) < (𝑒, 𝑓, 𝑛) 𝑇(𝑒, 𝑓) 𝐸(𝑒, 𝑓, 𝑛) 𝐹(𝑒, 𝑓, 𝑛) 𝐿(𝑒, 𝑓, 𝑛)

Proof of Theorem fldext2chn Dummy variables 𝑐 𝑑 𝑔 𝑚 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldext2chn.3	. . 3 ⊢ (𝜑 → 0 < (♯‘𝑇))
2		fveq2 6908	. . . . . 6 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
3	2	breq2d 5118	. . . . 5 ⊢ (𝑑 = ∅ → (0 < (♯‘𝑑) ↔ 0 < (♯‘∅)))
4		fveq2 6908	. . . . . . . 8 ⊢ (𝑑 = ∅ → (lastS‘𝑑) = (lastS‘∅))
5	4	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = ∅ → (𝑊 ↾_s (lastS‘𝑑)) = (𝑊 ↾_s (lastS‘∅)))
6		fveq1 6907	. . . . . . . 8 ⊢ (𝑑 = ∅ → (𝑑‘0) = (∅‘0))
7	6	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = ∅ → (𝑊 ↾_s (𝑑‘0)) = (𝑊 ↾_s (∅‘0)))
8	5, 7	breq12d 5119	. . . . . 6 ⊢ (𝑑 = ∅ → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ↔ (𝑊 ↾_s (lastS‘∅))/_FldExt(𝑊 ↾_s (∅‘0))))
9	5, 7	oveq12d 7461	. . . . . . . 8 ⊢ (𝑑 = ∅ → ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = ((𝑊 ↾_s (lastS‘∅))[:](𝑊 ↾_s (∅‘0))))
10	9	eqeq1d 2765	. . . . . . 7 ⊢ (𝑑 = ∅ → (((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ((𝑊 ↾_s (lastS‘∅))[:](𝑊 ↾_s (∅‘0))) = (2↑𝑛)))
11	10	rexbidv 3190	. . . . . 6 ⊢ (𝑑 = ∅ → (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘∅))[:](𝑊 ↾_s (∅‘0))) = (2↑𝑛)))
12	8, 11	anbi12d 644	. . . . 5 ⊢ (𝑑 = ∅ → (((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛)) ↔ ((𝑊 ↾_s (lastS‘∅))/_FldExt(𝑊 ↾_s (∅‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘∅))[:](𝑊 ↾_s (∅‘0))) = (2↑𝑛))))
13	3, 12	imbi12d 347	. . . 4 ⊢ (𝑑 = ∅ → ((0 < (♯‘𝑑) → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛))) ↔ (0 < (♯‘∅) → ((𝑊 ↾_s (lastS‘∅))/_FldExt(𝑊 ↾_s (∅‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘∅))[:](𝑊 ↾_s (∅‘0))) = (2↑𝑛)))))
14		fveq2 6908	. . . . . 6 ⊢ (𝑑 = 𝑐 → (♯‘𝑑) = (♯‘𝑐))
15	14	breq2d 5118	. . . . 5 ⊢ (𝑑 = 𝑐 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑐)))
16		fveq2 6908	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (lastS‘𝑑) = (lastS‘𝑐))
17	16	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (𝑊 ↾_s (lastS‘𝑑)) = (𝑊 ↾_s (lastS‘𝑐)))
18		fveq1 6907	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (𝑑‘0) = (𝑐‘0))
19	18	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (𝑊 ↾_s (𝑑‘0)) = (𝑊 ↾_s (𝑐‘0)))
20	17, 19	breq12d 5119	. . . . . 6 ⊢ (𝑑 = 𝑐 → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ↔ (𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0))))
21	17, 19	oveq12d 7461	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))))
22	21	eqeq1d 2765	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))
23	22	rexbidv 3190	. . . . . 6 ⊢ (𝑑 = 𝑐 → (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))
24	20, 23	anbi12d 644	. . . . 5 ⊢ (𝑑 = 𝑐 → (((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛)) ↔ ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛))))
25	15, 24	imbi12d 347	. . . 4 ⊢ (𝑑 = 𝑐 → ((0 < (♯‘𝑑) → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛))) ↔ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))))
26		fveq2 6908	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (♯‘𝑑) = (♯‘(𝑐 ++ ⟨“𝑔”⟩)))
27	26	breq2d 5118	. . . . 5 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (0 < (♯‘𝑑) ↔ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))))
28		fveq2 6908	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (lastS‘𝑑) = (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))
29	28	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (𝑊 ↾_s (lastS‘𝑑)) = (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩))))
30		fveq1 6907	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (𝑑‘0) = ((𝑐 ++ ⟨“𝑔”⟩)‘0))
31	30	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (𝑊 ↾_s (𝑑‘0)) = (𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)))
32	29, 31	breq12d 5119	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ↔ (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))))
33	29, 31	oveq12d 7461	. . . . . . . . 9 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))))
34	33	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑛)))
35	34	rexbidv 3190	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑛)))
36		oveq2 7451	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
37	36	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑛) ↔ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚)))
38	37	cbvrexvw 3245	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑛) ↔ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚))
39	35, 38	bitrdi 290	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚)))
40	32, 39	anbi12d 644	. . . . 5 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛)) ↔ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) ∧ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚))))
41	27, 40	imbi12d 347	. . . 4 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((0 < (♯‘𝑑) → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛))) ↔ (0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩)) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) ∧ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚)))))
42		fveq2 6908	. . . . . 6 ⊢ (𝑑 = 𝑇 → (♯‘𝑑) = (♯‘𝑇))
43	42	breq2d 5118	. . . . 5 ⊢ (𝑑 = 𝑇 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑇)))
44		fveq2 6908	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (lastS‘𝑑) = (lastS‘𝑇))
45	44	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (𝑊 ↾_s (lastS‘𝑑)) = (𝑊 ↾_s (lastS‘𝑇)))
46		fveq1 6907	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (𝑑‘0) = (𝑇‘0))
47	46	oveq2d 7459	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (𝑊 ↾_s (𝑑‘0)) = (𝑊 ↾_s (𝑇‘0)))
48	45, 47	breq12d 5119	. . . . . 6 ⊢ (𝑑 = 𝑇 → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ↔ (𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0))))
49	45, 47	oveq12d 7461	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))))
50	49	eqeq1d 2765	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛)))
51	50	rexbidv 3190	. . . . . 6 ⊢ (𝑑 = 𝑇 → (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛)))
52	48, 51	anbi12d 644	. . . . 5 ⊢ (𝑑 = 𝑇 → (((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛)) ↔ ((𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛))))
53	43, 52	imbi12d 347	. . . 4 ⊢ (𝑑 = 𝑇 → ((0 < (♯‘𝑑) → ((𝑊 ↾_s (lastS‘𝑑))/_FldExt(𝑊 ↾_s (𝑑‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑑))[:](𝑊 ↾_s (𝑑‘0))) = (2↑𝑛))) ↔ (0 < (♯‘𝑇) → ((𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛)))))
54		fldext2chn.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( < Chain (SubDRing‘𝑊)))
55		0re 11337	. . . . . . . 8 ⊢ 0 ∈ ℝ
56	55	ltnri 11446	. . . . . . 7 ⊢ ¬ 0 < 0
57	56	a1i 11	. . . . . 6 ⊢ (𝜑 → ¬ 0 < 0)
58		hash0 14538	. . . . . . 7 ⊢ (♯‘∅) = 0
59	58	breq2i 5114	. . . . . 6 ⊢ (0 < (♯‘∅) ↔ 0 < 0)
60	57, 59	sylnibr 332	. . . . 5 ⊢ (𝜑 → ¬ 0 < (♯‘∅))
61	60	pm2.21d 122	. . . 4 ⊢ (𝜑 → (0 < (♯‘∅) → ((𝑊 ↾_s (lastS‘∅))/_FldExt(𝑊 ↾_s (∅‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘∅))[:](𝑊 ↾_s (∅‘0))) = (2↑𝑛))))
62		fldext2chn.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ Field)
63	62	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑊 ∈ Field)
64		simp-5r 798	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑔 ∈ (SubDRing‘𝑊))
65		fldsdrgfld 21085	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Field ∧ 𝑔 ∈ (SubDRing‘𝑊)) → (𝑊 ↾_s 𝑔) ∈ Field)
66	63, 64, 65	syl2anc 596	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑊 ↾_s 𝑔) ∈ Field)
67		fldextid 34361	. . . . . . . . 9 ⊢ ((𝑊 ↾_s 𝑔) ∈ Field → (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s 𝑔))
68	66, 67	syl 18	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s 𝑔))
69		simp-5r 798	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) → 𝑐 ∈ ( < Chain (SubDRing‘𝑊)))
70	69	chnwrd 18811	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) → 𝑐 ∈ Word (SubDRing‘𝑊))
71	70	adantr 486	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑐 ∈ Word (SubDRing‘𝑊))
72		lswccats1 14810	. . . . . . . . . 10 ⊢ ((𝑐 ∈ Word (SubDRing‘𝑊) ∧ 𝑔 ∈ (SubDRing‘𝑊)) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
73	71, 64, 72	syl2anc 596	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
74	73	oveq2d 7459	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩))) = (𝑊 ↾_s 𝑔))
75		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑐 = ∅)
76	75	oveq1d 7458	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑐 ++ ⟨“𝑔”⟩) = (∅ ++ ⟨“𝑔”⟩))
77		s0s1 15102	. . . . . . . . . . . 12 ⊢ ⟨“𝑔”⟩ = (∅ ++ ⟨“𝑔”⟩)
78	76, 77	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑐 ++ ⟨“𝑔”⟩) = ⟨“𝑔”⟩)
79	78	fveq1d 6910	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (⟨“𝑔”⟩‘0))
80		s1fv 14786	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (SubDRing‘𝑊) → (⟨“𝑔”⟩‘0) = 𝑔)
81	64, 80	syl 18	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (⟨“𝑔”⟩‘0) = 𝑔)
82	79, 81	eqtrd 2798	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = 𝑔)
83	82	oveq2d 7459	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (𝑊 ↾_s 𝑔))
84	68, 74, 83	3brtr4d 5140	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)))
85		oveq2 7451	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
86		2cn 12445	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
87		exp0 14235	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑0) = 1)
88	86, 87	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑0) = 1
89	85, 88	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑚 = 0 → (2↑𝑚) = 1)
90	89	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑚 = 0 → (((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚) ↔ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = 1))
91		0nn0 12648	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
92	91	a1i 11	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 0 ∈ ℕ₀)
93	74, 83	oveq12d 7461	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s 𝑔)))
94		extdgid 34362	. . . . . . . . . 10 ⊢ ((𝑊 ↾_s 𝑔) ∈ Field → ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s 𝑔)) = 1)
95	66, 94	syl 18	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s 𝑔)) = 1)
96	93, 95	eqtrd 2798	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = 1)
97	90, 92, 96	rspcedvdw 3583	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚))
98	84, 97	jca 521	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) ∧ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚)))
99		simp-6r 800	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔))
100		simpllr 788	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 𝑐 ≠ ∅)
101	100	neneqd 2964	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ¬ 𝑐 = ∅)
102	99, 101	orcnd 892	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (lastS‘𝑐) < 𝑔)
103	70	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 𝑐 ∈ Word (SubDRing‘𝑊))
104		lswcl 14740	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ Word (SubDRing‘𝑊) ∧ 𝑐 ≠ ∅) → (lastS‘𝑐) ∈ (SubDRing‘𝑊))
105	103, 100, 104	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (lastS‘𝑐) ∈ (SubDRing‘𝑊))
106		simp-7r 802	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 𝑔 ∈ (SubDRing‘𝑊))
107		fldext2chn.e	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐸 = (𝑊 ↾_s 𝑒)
108		fldext2chn.f	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐹 = (𝑊 ↾_s 𝑓)
109	107, 108	breq12i 5115	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 ↔ (𝑊 ↾_s 𝑒)/_FldExt(𝑊 ↾_s 𝑓))
110	107, 108	oveq12i 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸[:]𝐹) = ((𝑊 ↾_s 𝑒)[:](𝑊 ↾_s 𝑓))
111	110	eqeq1i 2768	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸[:]𝐹) = 2 ↔ ((𝑊 ↾_s 𝑒)[:](𝑊 ↾_s 𝑓)) = 2)
112	109, 111	anbi12i 640	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2) ↔ ((𝑊 ↾_s 𝑒)/_FldExt(𝑊 ↾_s 𝑓) ∧ ((𝑊 ↾_s 𝑒)[:](𝑊 ↾_s 𝑓)) = 2))
113		oveq2 7451	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑔 → (𝑊 ↾_s 𝑒) = (𝑊 ↾_s 𝑔))
114	113	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑊 ↾_s 𝑒) = (𝑊 ↾_s 𝑔))
115		oveq2 7451	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (lastS‘𝑐) → (𝑊 ↾_s 𝑓) = (𝑊 ↾_s (lastS‘𝑐)))
116	115	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑊 ↾_s 𝑓) = (𝑊 ↾_s (lastS‘𝑐)))
117	114, 116	breq12d 5119	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑊 ↾_s 𝑒)/_FldExt(𝑊 ↾_s 𝑓) ↔ (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐))))
118	114, 116	oveq12d 7461	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑊 ↾_s 𝑒)[:](𝑊 ↾_s 𝑓)) = ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))))
119	118	eqeq1d 2765	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (((𝑊 ↾_s 𝑒)[:](𝑊 ↾_s 𝑓)) = 2 ↔ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2))
120	117, 119	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (((𝑊 ↾_s 𝑒)/_FldExt(𝑊 ↾_s 𝑓) ∧ ((𝑊 ↾_s 𝑒)[:](𝑊 ↾_s 𝑓)) = 2) ↔ ((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2)))
121	112, 120	bitrid 286	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2) ↔ ((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2)))
122	121	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (lastS‘𝑐) ∧ 𝑒 = 𝑔) → ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2) ↔ ((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2)))
123		fldext2chn.l	. . . . . . . . . . . . . 14 ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
124	122, 123	brabga 5512	. . . . . . . . . . . . 13 ⊢ (((lastS‘𝑐) ∈ (SubDRing‘𝑊) ∧ 𝑔 ∈ (SubDRing‘𝑊)) → ((lastS‘𝑐) < 𝑔 ↔ ((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2)))
125	105, 106, 124	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((lastS‘𝑐) < 𝑔 ↔ ((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2)))
126	102, 125	mpbid 235	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2))
127	126	simpld 500	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)))
128		hashgt0 14559	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ ( < Chain (SubDRing‘𝑊)) ∧ 𝑐 ≠ ∅) → 0 < (♯‘𝑐))
129	69, 128	sylan 592	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → 0 < (♯‘𝑐))
130		simpllr 788	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛))))
131	129, 130	mpd 16	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))
132	131	simprd 501	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛))
133		oveq2 7451	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑜 → (2↑𝑛) = (2↑𝑜))
134	133	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑜 → (((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛) ↔ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)))
135	134	cbvrexvw 3245	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛) ↔ ∃𝑜 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜))
136	132, 135	sylib 221	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ∃𝑜 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜))
137	127, 136	r19.29a 3174	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)))
138	131	simpld 500	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)))
139		fldexttr 34360	. . . . . . . . 9 ⊢ (((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ (𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0))) → (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (𝑐‘0)))
140	137, 138, 139	syl2anc 596	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (𝑐‘0)))
141	103, 106, 72	syl2anc 596	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
142	141	oveq2d 7459	. . . . . . . . 9 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩))) = (𝑊 ↾_s 𝑔))
143	142, 136	r19.29a 3174	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩))) = (𝑊 ↾_s 𝑔))
144	106	s1cld 14777	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ⟨“𝑔”⟩ ∈ Word (SubDRing‘𝑊))
145	129	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 0 < (♯‘𝑐))
146		ccatfv0 14756	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Word (SubDRing‘𝑊) ∧ ⟨“𝑔”⟩ ∈ Word (SubDRing‘𝑊) ∧ 0 < (♯‘𝑐)) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
147	103, 144, 145, 146	syl3anc 1399	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
148	147	oveq2d 7459	. . . . . . . . 9 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (𝑊 ↾_s (𝑐‘0)))
149	148, 136	r19.29a 3174	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (𝑊 ↾_s (𝑐‘0)))
150	140, 143, 149	3brtr4d 5140	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)))
151		oveq2 7451	. . . . . . . . . 10 ⊢ (𝑚 = (𝑜 + 1) → (2↑𝑚) = (2↑(𝑜 + 1)))
152	151	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑚 = (𝑜 + 1) → (((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚) ↔ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑(𝑜 + 1))))
153		simplr 781	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 𝑜 ∈ ℕ₀)
154		1nn0 12649	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ₀
155	154	a1i 11	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 1 ∈ ℕ₀)
156	153, 155	nn0addcld 12698	. . . . . . . . 9 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (𝑜 + 1) ∈ ℕ₀)
157	142, 148	oveq12d 7461	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (𝑐‘0))))
158	138	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)))
159		extdgmul 34365	. . . . . . . . . . 11 ⊢ (((𝑊 ↾_s 𝑔)/_FldExt(𝑊 ↾_s (lastS‘𝑐)) ∧ (𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0))) → ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (𝑐‘0))) = (((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) ·_e ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0)))))
160	127, 158, 159	syl2anc 596	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (𝑐‘0))) = (((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) ·_e ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0)))))
161	86	a1i 11	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 2 ∈ ℂ)
162	161, 153	expcld 14316	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (2↑𝑜) ∈ ℂ)
163	161, 162	mulcomd 11357	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (2 · (2↑𝑜)) = ((2↑𝑜) · 2))
164	126	simprd 501	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) = 2)
165		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜))
166	164, 165	oveq12d 7461	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) ·_e ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0)))) = (2 ·_e (2↑𝑜)))
167		2re 12444	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
168	167	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → 2 ∈ ℝ)
169	168, 153	reexpcld 14333	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (2↑𝑜) ∈ ℝ)
170		rexmul 13428	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ (2↑𝑜) ∈ ℝ) → (2 ·_e (2↑𝑜)) = (2 · (2↑𝑜)))
171	168, 169, 170	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (2 ·_e (2↑𝑜)) = (2 · (2↑𝑜)))
172	166, 171	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) ·_e ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0)))) = (2 · (2↑𝑜)))
173	161, 153	expp1d 14317	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (2↑(𝑜 + 1)) = ((2↑𝑜) · 2))
174	163, 172, 173	3eqtr4d 2808	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → (((𝑊 ↾_s 𝑔)[:](𝑊 ↾_s (lastS‘𝑐))) ·_e ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0)))) = (2↑(𝑜 + 1)))
175	157, 160, 174	3eqtrd 2802	. . . . . . . . 9 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑(𝑜 + 1)))
176	152, 156, 175	rspcedvdw 3583	. . . . . . . 8 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑜)) → ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚))
177	176, 136	r19.29a 3174	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚))
178	150, 177	jca 521	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) ∧ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚)))
179	98, 178	pm2.61dane 3046	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) ∧ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚)))
180	179	ex 418	. . . 4 ⊢ (((((𝜑 ∧ 𝑐 ∈ ( < Chain (SubDRing‘𝑊))) ∧ 𝑔 ∈ (SubDRing‘𝑊)) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((𝑊 ↾_s (lastS‘𝑐))/_FldExt(𝑊 ↾_s (𝑐‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑐))[:](𝑊 ↾_s (𝑐‘0))) = (2↑𝑛)))) → (0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩)) → ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))/_FldExt(𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0)) ∧ ∃𝑚 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))[:](𝑊 ↾_s ((𝑐 ++ ⟨“𝑔”⟩)‘0))) = (2↑𝑚))))
181	13, 25, 41, 53, 54, 61, 180	chnind 18824	. . 3 ⊢ (𝜑 → (0 < (♯‘𝑇) → ((𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛))))
182	1, 181	mpd 16	. 2 ⊢ (𝜑 → ((𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛)))
183		fldext2chn.2	. . . 4 ⊢ (𝜑 → (𝑊 ↾_s (lastS‘𝑇)) = 𝐿)
184		fldext2chn.1	. . . 4 ⊢ (𝜑 → (𝑊 ↾_s (𝑇‘0)) = 𝑄)
185	183, 184	breq12d 5119	. . 3 ⊢ (𝜑 → ((𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0)) ↔ 𝐿/_FldExt𝑄))
186	183, 184	oveq12d 7461	. . . . 5 ⊢ (𝜑 → ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (𝐿[:]𝑄))
187	186	eqeq1d 2765	. . . 4 ⊢ (𝜑 → (((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛) ↔ (𝐿[:]𝑄) = (2↑𝑛)))
188	187	rexbidv 3190	. . 3 ⊢ (𝜑 → (∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ (𝐿[:]𝑄) = (2↑𝑛)))
189	185, 188	anbi12d 644	. 2 ⊢ (𝜑 → (((𝑊 ↾_s (lastS‘𝑇))/_FldExt(𝑊 ↾_s (𝑇‘0)) ∧ ∃𝑛 ∈ ℕ₀ ((𝑊 ↾_s (lastS‘𝑇))[:](𝑊 ↾_s (𝑇‘0))) = (2↑𝑛)) ↔ (𝐿/_FldExt𝑄 ∧ ∃𝑛 ∈ ℕ₀ (𝐿[:]𝑄) = (2↑𝑛))))
190	182, 189	mpbid 235	1 ⊢ (𝜑 → (𝐿/_FldExt𝑄 ∧ ∃𝑛 ∈ ℕ₀ (𝐿[:]𝑄) = (2↑𝑛)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 ∅c0 4281 class class class wbr 5106 {copab 5170 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 · cmul 11232 < clt 11370 2c2 12424 ℕ₀cn0 12633 ·_e cxmu 13267 ↑cexp 14231 ♯chash 14501 Word cword 14685 lastSclsw 14734 ++ cconcat 14742 ⟨“cs1 14769 ↾_s cress 17437 Chain cchn 18808 Fieldcfield 21011 SubDRingcsdrg 21073 /_FldExtcfldext 34340 [:]cextdg 34342

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-ac2 10568 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-rpss 7763 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8745 df-map 8877 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-sup 9461 df-oi 9531 df-dju 10009 df-card 10047 df-acn 10050 df-ac 10222 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-xnn0 12707 df-z 12721 df-dec 12842 df-uz 12993 df-rp 13148 df-xmul 13270 df-fz 13667 df-fzo 13816 df-seq 14172 df-exp 14232 df-hash 14502 df-word 14686 df-lsw 14735 df-concat 14743 df-s1 14770 df-substr 14817 df-pfx 14849 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ocomp 17478 df-ds 17479 df-hom 17481 df-cco 17482 df-0g 17641 df-gsum 17642 df-prds 17647 df-pws 17649 df-mre 17785 df-mrc 17786 df-mri 17787 df-acs 17788 df-proset 18497 df-drs 18498 df-poset 18516 df-ipo 18731 df-chn 18809 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-ghm 19457 df-cntz 19561 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-nzr 20793 df-subrng 20828 df-subrg 20852 df-drng 21012 df-field 21013 df-sdrg 21074 df-lmod 21167 df-lss 21237 df-lsp 21277 df-lmhm 21327 df-lbs 21380 df-lvec 21408 df-sra 21478 df-rgmod 21479 df-lidl 21516 df-rsp 21517 df-dsmm 22068 df-frlm 22083 df-uvc 22119 df-lindf 22142 df-linds 22143 df-dim 34302 df-fldext 34343 df-extdg 34344

This theorem is used by: constrext2chnlem 34452

W3C validator