fldext2chn - Mathbox for Thierry Arnoux

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Theorem fldext2chn 33601

Description: In a non-empty tower 𝑇 of quadratic field extensions, the degree of the extension of the first member by the last is a power of two. (Contributed by Thierry Arnoux, 19-Jun-2025.)

**Hypotheses**
Ref	Expression
fldext2chn.l	⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) = 2)}
fldext2chn.t	⊢ (𝜑 → 𝑇 ∈ ( < ChainField))
fldext2chn.1	⊢ (𝜑 → (𝑇‘0) = 𝑄)
fldext2chn.2	⊢ (𝜑 → (lastS‘𝑇) = 𝐹)
fldext2chn.3	⊢ (𝜑 → 0 < (♯‘𝑇))

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

Distinct variable groups: 𝑇,𝑛 𝜑,𝑛 𝑒,𝑓 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝑄(𝑒,𝑓,𝑛) < (𝑒,𝑓,𝑛) 𝑇(𝑒,𝑓) 𝐹(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	. . . 4 ⊢ (𝜑 → 0 < (♯‘𝑇))
2		fveq2 6893	. . . . . . 7 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
3	2	breq2d 5157	. . . . . 6 ⊢ (𝑑 = ∅ → (0 < (♯‘𝑑) ↔ 0 < (♯‘∅)))
4		fveq2 6893	. . . . . . . 8 ⊢ (𝑑 = ∅ → (lastS‘𝑑) = (lastS‘∅))
5		fveq1 6892	. . . . . . . 8 ⊢ (𝑑 = ∅ → (𝑑‘0) = (∅‘0))
6	4, 5	breq12d 5158	. . . . . . 7 ⊢ (𝑑 = ∅ → ((lastS‘𝑑)/_FldExt(𝑑‘0) ↔ (lastS‘∅)/_FldExt(∅‘0)))
7	4, 5	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑑 = ∅ → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘∅)[:](∅‘0)))
8	7	eqeq1d 2728	. . . . . . . 8 ⊢ (𝑑 = ∅ → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))
9	8	rexbidv 3169	. . . . . . 7 ⊢ (𝑑 = ∅ → (∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))
10	6, 9	anbi12d 630	. . . . . 6 ⊢ (𝑑 = ∅ → (((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛)) ↔ ((lastS‘∅)/_FldExt(∅‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛))))
11	3, 10	imbi12d 343	. . . . 5 ⊢ (𝑑 = ∅ → ((0 < (♯‘𝑑) → ((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛))) ↔ (0 < (♯‘∅) → ((lastS‘∅)/_FldExt(∅‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))))
12		fveq2 6893	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (♯‘𝑑) = (♯‘𝑐))
13	12	breq2d 5157	. . . . . 6 ⊢ (𝑑 = 𝑐 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑐)))
14		fveq2 6893	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (lastS‘𝑑) = (lastS‘𝑐))
15		fveq1 6892	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (𝑑‘0) = (𝑐‘0))
16	14, 15	breq12d 5158	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((lastS‘𝑑)/_FldExt(𝑑‘0) ↔ (lastS‘𝑐)/_FldExt(𝑐‘0)))
17	14, 15	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘𝑐)[:](𝑐‘0)))
18	17	eqeq1d 2728	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
19	18	rexbidv 3169	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
20	16, 19	anbi12d 630	. . . . . 6 ⊢ (𝑑 = 𝑐 → (((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛)) ↔ ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛))))
21	13, 20	imbi12d 343	. . . . 5 ⊢ (𝑑 = 𝑐 → ((0 < (♯‘𝑑) → ((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛))) ↔ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))))
22		fveq2 6893	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (♯‘𝑑) = (♯‘(𝑐 ++ ⟨“𝑔”⟩)))
23	22	breq2d 5157	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (0 < (♯‘𝑑) ↔ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))))
24		fveq2 6893	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (lastS‘𝑑) = (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))
25		fveq1 6892	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (𝑑‘0) = ((𝑐 ++ ⟨“𝑔”⟩)‘0))
26	24, 25	breq12d 5158	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((lastS‘𝑑)/_FldExt(𝑑‘0) ↔ (lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0)))
27	24, 25	oveq12d 7434	. . . . . . . . . 10 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)))
28	27	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛)))
29	28	rexbidv 3169	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛)))
30		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
31	30	eqeq2d 2737	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
32	31	cbvrexvw 3226	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛) ↔ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚))
33	29, 32	bitrdi 286	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
34	26, 33	anbi12d 630	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛)) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0) ∧ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚))))
35	23, 34	imbi12d 343	. . . . 5 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((0 < (♯‘𝑑) → ((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛))) ↔ (0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩)) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0) ∧ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))))
36		fveq2 6893	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (♯‘𝑑) = (♯‘𝑇))
37	36	breq2d 5157	. . . . . 6 ⊢ (𝑑 = 𝑇 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑇)))
38		fveq2 6893	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (lastS‘𝑑) = (lastS‘𝑇))
39		fveq1 6892	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (𝑑‘0) = (𝑇‘0))
40	38, 39	breq12d 5158	. . . . . . 7 ⊢ (𝑑 = 𝑇 → ((lastS‘𝑑)/_FldExt(𝑑‘0) ↔ (lastS‘𝑇)/_FldExt(𝑇‘0)))
41	38, 39	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑑 = 𝑇 → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘𝑇)[:](𝑇‘0)))
42	41	eqeq1d 2728	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
43	42	rexbidv 3169	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
44	40, 43	anbi12d 630	. . . . . 6 ⊢ (𝑑 = 𝑇 → (((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛)) ↔ ((lastS‘𝑇)/_FldExt(𝑇‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛))))
45	37, 44	imbi12d 343	. . . . 5 ⊢ (𝑑 = 𝑇 → ((0 < (♯‘𝑑) → ((lastS‘𝑑)/_FldExt(𝑑‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛))) ↔ (0 < (♯‘𝑇) → ((lastS‘𝑇)/_FldExt(𝑇‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))))
46		fldext2chn.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ( < ChainField))
47		0re 11257	. . . . . . . . 9 ⊢ 0 ∈ ℝ
48	47	ltnri 11364	. . . . . . . 8 ⊢ ¬ 0 < 0
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → ¬ 0 < 0)
50		hash0 14379	. . . . . . . 8 ⊢ (♯‘∅) = 0
51	50	breq2i 5153	. . . . . . 7 ⊢ (0 < (♯‘∅) ↔ 0 < 0)
52	49, 51	sylnibr 328	. . . . . 6 ⊢ (𝜑 → ¬ 0 < (♯‘∅))
53	52	pm2.21d 121	. . . . 5 ⊢ (𝜑 → (0 < (♯‘∅) → ((lastS‘∅)/_FldExt(∅‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛))))
54		simp-5r 784	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑔 ∈ Field)
55		fldextid 33554	. . . . . . . . . 10 ⊢ (𝑔 ∈ Field → 𝑔/_FldExt𝑔)
56	54, 55	syl 17	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑔/_FldExt𝑔)
57		simp-5r 784	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) → 𝑐 ∈ ( < ChainField))
58	57	chnwrd 32880	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) → 𝑐 ∈ Word Field)
59		lswccats1 14637	. . . . . . . . . 10 ⊢ ((𝑐 ∈ Word Field ∧ 𝑔 ∈ Field) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
60	58, 54, 59	syl2an2r 683	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
61		simpr 483	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 𝑐 = ∅)
62	61	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑐 ++ ⟨“𝑔”⟩) = (∅ ++ ⟨“𝑔”⟩))
63		s0s1 14926	. . . . . . . . . . . 12 ⊢ ⟨“𝑔”⟩ = (∅ ++ ⟨“𝑔”⟩)
64	62, 63	eqtr4di 2784	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑐 ++ ⟨“𝑔”⟩) = ⟨“𝑔”⟩)
65	64	fveq1d 6895	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (⟨“𝑔”⟩‘0))
66		s1fv 14613	. . . . . . . . . . 11 ⊢ (𝑔 ∈ Field → (⟨“𝑔”⟩‘0) = 𝑔)
67	54, 66	syl 17	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (⟨“𝑔”⟩‘0) = 𝑔)
68	65, 67	eqtrd 2766	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = 𝑔)
69	56, 60, 68	3brtr4d 5177	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0))
70		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
71		2cn 12333	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
72		exp0 14079	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ → (2↑0) = 1)
73	71, 72	ax-mp 5	. . . . . . . . . . 11 ⊢ (2↑0) = 1
74	70, 73	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (2↑𝑚) = 1)
75	74	eqeq2d 2737	. . . . . . . . 9 ⊢ (𝑚 = 0 → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = 1))
76		0nn0 12533	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
77	76	a1i 11	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → 0 ∈ ℕ₀)
78	60, 68	oveq12d 7434	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (𝑔[:]𝑔))
79		extdgid 33555	. . . . . . . . . . 11 ⊢ (𝑔 ∈ Field → (𝑔[:]𝑔) = 1)
80	54, 79	syl 17	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → (𝑔[:]𝑔) = 1)
81	78, 80	eqtrd 2766	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = 1)
82	75, 77, 81	rspcedvdw 3610	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚))
83	69, 82	jca 510	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 = ∅) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0) ∧ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
84		simp-6r 786	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔))
85		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 𝑐 ≠ ∅)
86	85	neneqd 2935	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ¬ 𝑐 = ∅)
87	84, 86	orcnd 876	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (lastS‘𝑐) < 𝑔)
88	58	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 𝑐 ∈ Word Field)
89		lswcl 14571	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ Word Field ∧ 𝑐 ≠ ∅) → (lastS‘𝑐) ∈ Field)
90	88, 85, 89	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (lastS‘𝑐) ∈ Field)
91		simp-7r 788	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 𝑔 ∈ Field)
92		breq12 5150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑒/_FldExt𝑓 ↔ 𝑔/_FldExt(lastS‘𝑐)))
93		oveq12 7425	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑒[:]𝑓) = (𝑔[:](lastS‘𝑐)))
94	93	eqeq1d 2728	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑒[:]𝑓) = 2 ↔ (𝑔[:](lastS‘𝑐)) = 2))
95	92, 94	anbi12d 630	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) = 2) ↔ (𝑔/_FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
96	95	ancoms 457	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (lastS‘𝑐) ∧ 𝑒 = 𝑔) → ((𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) = 2) ↔ (𝑔/_FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
97		fldext2chn.l	. . . . . . . . . . . . . . 15 ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) = 2)}
98	96, 97	brabga 5532	. . . . . . . . . . . . . 14 ⊢ (((lastS‘𝑐) ∈ Field ∧ 𝑔 ∈ Field) → ((lastS‘𝑐) < 𝑔 ↔ (𝑔/_FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
99	90, 91, 98	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((lastS‘𝑐) < 𝑔 ↔ (𝑔/_FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
100	87, 99	mpbid 231	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (𝑔/_FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2))
101	100	simpld 493	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 𝑔/_FldExt(lastS‘𝑐))
102		hashgt0 14400	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ( < ChainField) ∧ 𝑐 ≠ ∅) → 0 < (♯‘𝑐))
103	57, 102	sylan 578	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → 0 < (♯‘𝑐))
104		simpllr 774	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛))))
105	103, 104	mpd 15	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
106	105	simprd 494	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛))
107		oveq2 7424	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑜 → (2↑𝑛) = (2↑𝑜))
108	107	eqeq2d 2737	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑜 → (((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛) ↔ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)))
109	108	cbvrexvw 3226	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛) ↔ ∃𝑜 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜))
110	106, 109	sylib 217	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ∃𝑜 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜))
111	101, 110	r19.29a 3152	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → 𝑔/_FldExt(lastS‘𝑐))
112	105	simpld 493	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (lastS‘𝑐)/_FldExt(𝑐‘0))
113		fldexttr 33553	. . . . . . . . . 10 ⊢ ((𝑔/_FldExt(lastS‘𝑐) ∧ (lastS‘𝑐)/_FldExt(𝑐‘0)) → 𝑔/_FldExt(𝑐‘0))
114	111, 112, 113	syl2anc 582	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → 𝑔/_FldExt(𝑐‘0))
115	88, 91, 59	syl2anc 582	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
116	115, 110	r19.29a 3152	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
117	91	s1cld 14606	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ⟨“𝑔”⟩ ∈ Word Field)
118	103	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 0 < (♯‘𝑐))
119		ccatfv0 14586	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Word Field ∧ ⟨“𝑔”⟩ ∈ Word Field ∧ 0 < (♯‘𝑐)) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
120	88, 117, 118, 119	syl3anc 1368	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
121	120, 110	r19.29a 3152	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
122	114, 116, 121	3brtr4d 5177	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0))
123		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑜 + 1) → (2↑𝑚) = (2↑(𝑜 + 1)))
124	123	eqeq2d 2737	. . . . . . . . . 10 ⊢ (𝑚 = (𝑜 + 1) → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑(𝑜 + 1))))
125		simplr 767	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 𝑜 ∈ ℕ₀)
126		1nn0 12534	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
127	126	a1i 11	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 1 ∈ ℕ₀)
128	125, 127	nn0addcld 12582	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (𝑜 + 1) ∈ ℕ₀)
129	115, 120	oveq12d 7434	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (𝑔[:](𝑐‘0)))
130	112	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (lastS‘𝑐)/_FldExt(𝑐‘0))
131		extdgmul 33556	. . . . . . . . . . . 12 ⊢ ((𝑔/_FldExt(lastS‘𝑐) ∧ (lastS‘𝑐)/_FldExt(𝑐‘0)) → (𝑔[:](𝑐‘0)) = ((𝑔[:](lastS‘𝑐)) ·_e ((lastS‘𝑐)[:](𝑐‘0))))
132	101, 130, 131	syl2anc 582	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (𝑔[:](𝑐‘0)) = ((𝑔[:](lastS‘𝑐)) ·_e ((lastS‘𝑐)[:](𝑐‘0))))
133		2cnd 12336	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 2 ∈ ℂ)
134	133, 125	expcld 14159	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (2↑𝑜) ∈ ℂ)
135	133, 134	mulcomd 11276	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (2 · (2↑𝑜)) = ((2↑𝑜) · 2))
136	100	simprd 494	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (𝑔[:](lastS‘𝑐)) = 2)
137		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜))
138	136, 137	oveq12d 7434	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((𝑔[:](lastS‘𝑐)) ·_e ((lastS‘𝑐)[:](𝑐‘0))) = (2 ·_e (2↑𝑜)))
139		2re 12332	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
140	139	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → 2 ∈ ℝ)
141	140, 125	reexpcld 14176	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (2↑𝑜) ∈ ℝ)
142		rexmul 13298	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (2↑𝑜) ∈ ℝ) → (2 ·_e (2↑𝑜)) = (2 · (2↑𝑜)))
143	140, 141, 142	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (2 ·_e (2↑𝑜)) = (2 · (2↑𝑜)))
144	138, 143	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((𝑔[:](lastS‘𝑐)) ·_e ((lastS‘𝑐)[:](𝑐‘0))) = (2 · (2↑𝑜)))
145	133, 125	expp1d 14160	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → (2↑(𝑜 + 1)) = ((2↑𝑜) · 2))
146	135, 144, 145	3eqtr4d 2776	. . . . . . . . . . 11 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((𝑔[:](lastS‘𝑐)) ·_e ((lastS‘𝑐)[:](𝑐‘0))) = (2↑(𝑜 + 1)))
147	129, 132, 146	3eqtrd 2770	. . . . . . . . . 10 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑(𝑜 + 1)))
148	124, 128, 147	rspcedvdw 3610	. . . . . . . . 9 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) ∧ 𝑜 ∈ ℕ₀) ∧ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)) → ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚))
149	148, 110	r19.29a 3152	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚))
150	122, 149	jca 510	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) ∧ 𝑐 ≠ ∅) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0) ∧ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
151	83, 150	pm2.61dane 3019	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) ∧ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0) ∧ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
152	151	ex 411	. . . . 5 ⊢ (((((𝜑 ∧ 𝑐 ∈ ( < ChainField)) ∧ 𝑔 ∈ Field) ∧ (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔)) ∧ (0 < (♯‘𝑐) → ((lastS‘𝑐)/_FldExt(𝑐‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))) → (0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩)) → ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))/_FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0) ∧ ∃𝑚 ∈ ℕ₀ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚))))
153	11, 21, 35, 45, 46, 53, 152	chnind 32883	. . . 4 ⊢ (𝜑 → (0 < (♯‘𝑇) → ((lastS‘𝑇)/_FldExt(𝑇‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛))))
154	1, 153	mpd 15	. . 3 ⊢ (𝜑 → ((lastS‘𝑇)/_FldExt(𝑇‘0) ∧ ∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
155	154	simprd 494	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛))
156		fldext2chn.2	. . . . 5 ⊢ (𝜑 → (lastS‘𝑇) = 𝐹)
157		fldext2chn.1	. . . . 5 ⊢ (𝜑 → (𝑇‘0) = 𝑄)
158	156, 157	oveq12d 7434	. . . 4 ⊢ (𝜑 → ((lastS‘𝑇)[:](𝑇‘0)) = (𝐹[:]𝑄))
159	158	eqeq1d 2728	. . 3 ⊢ (𝜑 → (((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛) ↔ (𝐹[:]𝑄) = (2↑𝑛)))
160	159	rexbidv 3169	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ₀ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ₀ (𝐹[:]𝑄) = (2↑𝑛)))
161	155, 160	mpbid 231	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐹[:]𝑄) = (2↑𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ∅c0 4322 class class class wbr 5145 {copab 5207 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 < clt 11289 2c2 12313 ℕ₀cn0 12518 ·_e cxmu 13139 ↑cexp 14075 ♯chash 14342 Word cword 14517 lastSclsw 14565 ++ cconcat 14573 ⟨“cs1 14598 Fieldcfield 20704 Chaincchn 32877 /_FldExtcfldext 33533 [:]cextdg 33536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-reg 9628 ax-inf2 9677 ax-ac2 10497 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-rpss 7726 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8726 df-map 8849 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-sup 9478 df-oi 9546 df-r1 9800 df-rank 9801 df-dju 9937 df-card 9975 df-acn 9978 df-ac 10152 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-xnn0 12591 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-xmul 13142 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-word 14518 df-lsw 14566 df-concat 14574 df-s1 14599 df-substr 14644 df-pfx 14674 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ocomp 17282 df-ds 17283 df-hom 17285 df-cco 17286 df-0g 17451 df-gsum 17452 df-prds 17457 df-pws 17459 df-mre 17594 df-mrc 17595 df-mri 17596 df-acs 17597 df-proset 18315 df-drs 18316 df-poset 18333 df-ipo 18548 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-cntz 19307 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-nzr 20491 df-subrng 20524 df-subrg 20549 df-drng 20705 df-field 20706 df-lmod 20834 df-lss 20905 df-lsp 20945 df-lmhm 20996 df-lbs 21049 df-lvec 21077 df-sra 21147 df-rgmod 21148 df-lidl 21193 df-rsp 21194 df-dsmm 21726 df-frlm 21741 df-uvc 21777 df-lindf 21800 df-linds 21801 df-chn 32878 df-dim 33500 df-fldext 33537 df-extdg 33538

This theorem is referenced by: (None)

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