Theorem fldext2chn 33734

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 6907	. . . . . . 7 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
3	2	breq2d 5160	. . . . . 6 ⊢ (𝑑 = ∅ → (0 < (♯‘𝑑) ↔ 0 < (♯‘∅)))
4		fveq2 6907	. . . . . . . 8 ⊢ (𝑑 = ∅ → (lastS‘𝑑) = (lastS‘∅))
5		fveq1 6906	. . . . . . . 8 ⊢ (𝑑 = ∅ → (𝑑‘0) = (∅‘0))
6	4, 5	breq12d 5161	. . . . . . 7 ⊢ (𝑑 = ∅ → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘∅)/FldExt(∅‘0)))
7	4, 5	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑑 = ∅ → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘∅)[:](∅‘0)))
8	7	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑑 = ∅ → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))
9	8	rexbidv 3177	. . . . . . 7 ⊢ (𝑑 = ∅ → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))
10	6, 9	anbi12d 632	. . . . . 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 344	. . . . 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 6907	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (♯‘𝑑) = (♯‘𝑐))
13	12	breq2d 5160	. . . . . 6 ⊢ (𝑑 = 𝑐 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑐)))
14		fveq2 6907	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (lastS‘𝑑) = (lastS‘𝑐))
15		fveq1 6906	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (𝑑‘0) = (𝑐‘0))
16	14, 15	breq12d 5161	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘𝑐)/FldExt(𝑐‘0)))
17	14, 15	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘𝑐)[:](𝑐‘0)))
18	17	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
19	18	rexbidv 3177	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
20	16, 19	anbi12d 632	. . . . . 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 344	. . . . 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 6907	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (♯‘𝑑) = (♯‘(𝑐 ++ ⟨“𝑔”⟩)))
23	22	breq2d 5160	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (0 < (♯‘𝑑) ↔ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))))
24		fveq2 6907	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (lastS‘𝑑) = (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))
25		fveq1 6906	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (𝑑‘0) = ((𝑐 ++ ⟨“𝑔”⟩)‘0))
26	24, 25	breq12d 5161	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘(𝑐 ++ ⟨“𝑔”⟩))/FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0)))
27	24, 25	oveq12d 7449	. . . . . . . . . 10 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)))
28	27	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛)))
29	28	rexbidv 3177	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛)))
30		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
31	30	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
32	31	cbvrexvw 3236	. . . . . . . 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 287	. . . . . . 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 632	. . . . . 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 344	. . . . 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 6907	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (♯‘𝑑) = (♯‘𝑇))
37	36	breq2d 5160	. . . . . 6 ⊢ (𝑑 = 𝑇 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑇)))
38		fveq2 6907	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (lastS‘𝑑) = (lastS‘𝑇))
39		fveq1 6906	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (𝑑‘0) = (𝑇‘0))
40	38, 39	breq12d 5161	. . . . . . 7 ⊢ (𝑑 = 𝑇 → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘𝑇)/FldExt(𝑇‘0)))
41	38, 39	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑑 = 𝑇 → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘𝑇)[:](𝑇‘0)))
42	41	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
43	42	rexbidv 3177	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
44	40, 43	anbi12d 632	. . . . . 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 344	. . . . 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 11261	. . . . . . . . 9 ⊢ 0 ∈ ℝ
48	47	ltnri 11368	. . . . . . . 8 ⊢ ¬ 0 < 0
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → ¬ 0 < 0)
50		hash0 14403	. . . . . . . 8 ⊢ (♯‘∅) = 0
51	50	breq2i 5156	. . . . . . 7 ⊢ (0 < (♯‘∅) ↔ 0 < 0)
52	49, 51	sylnibr 329	. . . . . 6 ⊢ (𝜑 → ¬ 0 < (♯‘∅))
53	52	pm2.21d 121	. . . . 5 ⊢ (𝜑 → (0 < (♯‘∅) → ((lastS‘∅)/FldExt(∅‘0) ∧ ∃𝑛 ∈ ℕ0 ((lastS‘∅)[:](∅‘0)) = (2↑𝑛))))
54		simp-5r 786	. . . . . . . . . 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 33687	. . . . . . . . . 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 786	. . . . . . . . . . 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 32982	. . . . . . . . . 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 14669	. . . . . . . . . 10 ⊢ ((𝑐 ∈ Word Field ∧ 𝑔 ∈ Field) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
60	58, 54, 59	syl2an2r 685	. . . . . . . . 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 484	. . . . . . . . . . . . 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 7446	. . . . . . . . . . . 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 14958	. . . . . . . . . . . 12 ⊢ ⟨“𝑔”⟩ = (∅ ++ ⟨“𝑔”⟩)
64	62, 63	eqtr4di 2793	. . . . . . . . . . 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 6909	. . . . . . . . . 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 14645	. . . . . . . . . . 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 2775	. . . . . . . . 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 5180	. . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
71		2cn 12339	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
72		exp0 14103	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ → (2↑0) = 1)
73	71, 72	ax-mp 5	. . . . . . . . . . 11 ⊢ (2↑0) = 1
74	70, 73	eqtrdi 2791	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (2↑𝑚) = 1)
75	74	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑚 = 0 → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = 1))
76		0nn0 12539	. . . . . . . . . 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 7449	. . . . . . . . . 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 33688	. . . . . . . . . . 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 2775	. . . . . . . . 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 3625	. . . . . . . 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 511	. . . . . . 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 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)) → (𝑐 = ∅ ∨ (lastS‘𝑐) < 𝑔))
85		simpllr 776	. . . . . . . . . . . . . . 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 2943	. . . . . . . . . . . . . 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 878	. . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 14603	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ Word Field ∧ 𝑐 ≠ ∅) → (lastS‘𝑐) ∈ Field)
90	88, 85, 89	syl2anc 584	. . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . 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 5153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑒/FldExt𝑓 ↔ 𝑔/FldExt(lastS‘𝑐)))
93		oveq12 7440	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑒[:]𝑓) = (𝑔[:](lastS‘𝑐)))
94	93	eqeq1d 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑒[:]𝑓) = 2 ↔ (𝑔[:](lastS‘𝑐)) = 2))
95	92, 94	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) = 2) ↔ (𝑔/FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
96	95	ancoms 458	. . . . . . . . . . . . . . 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 5544	. . . . . . . . . . . . . 14 ⊢ (((lastS‘𝑐) ∈ Field ∧ 𝑔 ∈ Field) → ((lastS‘𝑐) < 𝑔 ↔ (𝑔/FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
99	90, 91, 98	syl2anc 584	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 14424	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ( < ChainField) ∧ 𝑐 ≠ ∅) → 0 < (♯‘𝑐))
103	57, 102	sylan 580	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑜 → (2↑𝑛) = (2↑𝑜))
108	107	eqeq2d 2746	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑜 → (((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛) ↔ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)))
109	108	cbvrexvw 3236	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕ0 ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛) ↔ ∃𝑜 ∈ ℕ0 ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜))
110	106, 109	sylib 218	. . . . . . . . . . 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 3160	. . . . . . . . . 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 494	. . . . . . . . . 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 33686	. . . . . . . . . 10 ⊢ ((𝑔/FldExt(lastS‘𝑐) ∧ (lastS‘𝑐)/FldExt(𝑐‘0)) → 𝑔/FldExt(𝑐‘0))
114	111, 112, 113	syl2anc 584	. . . . . . . . 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 584	. . . . . . . . . 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 3160	. . . . . . . . 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 14638	. . . . . . . . . . 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 726	. . . . . . . . . . 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 14618	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Word Field ∧ ⟨“𝑔”⟩ ∈ Word Field ∧ 0 < (♯‘𝑐)) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
120	88, 117, 118, 119	syl3anc 1370	. . . . . . . . . 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 3160	. . . . . . . . 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 5180	. . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑜 + 1) → (2↑𝑚) = (2↑(𝑜 + 1)))
124	123	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑚 = (𝑜 + 1) → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑(𝑜 + 1))))
125		simplr 769	. . . . . . . . . . 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 12540	. . . . . . . . . . . 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 12589	. . . . . . . . . 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 7449	. . . . . . . . . . 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 726	. . . . . . . . . . . 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 33689	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 12342	. . . . . . . . . . . . 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 14183	. . . . . . . . . . . . 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 11280	. . . . . . . . . . . 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 495	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . 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 12338	. . . . . . . . . . . . . . 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 14200	. . . . . . . . . . . . . 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 13310	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (2↑𝑜) ∈ ℝ) → (2 ·e (2↑𝑜)) = (2 · (2↑𝑜)))
143	140, 141, 142	syl2anc 584	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 14184	. . . . . . . . . . . 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 2785	. . . . . . . . . . 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 2779	. . . . . . . . . 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 3625	. . . . . . . . 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 3160	. . . . . . . 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 511	. . . . . . 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 3027	. . . . . 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 412	. . . . 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 32985	. . . 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 495	. 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 7449	. . . 4 ⊢ (𝜑 → ((lastS‘𝑇)[:](𝑇‘0)) = (𝐹[:]𝑄))
159	158	eqeq1d 2737	. . 3 ⊢ (𝜑 → (((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛) ↔ (𝐹[:]𝑄) = (2↑𝑛)))
160	159	rexbidv 3177	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ0 ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 (𝐹[:]𝑄) = (2↑𝑛)))
161	155, 160	mpbid 232	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐹[:]𝑄) = (2↑𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  ∅c0 4339   class class class wbr 5148  {copab 5210  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  2c2 12319  ℕ₀cn0 12524   ·_e cxmu 13151  ↑cexp 14099  ♯chash 14366  Word cword 14549  lastSclsw 14597   ++ cconcat 14605  ⟨“cs1 14630  Fieldcfield 20747  Chaincchn 32979  /_FldExtcfldext 33666  [:]cextdg 33669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-xmul 13154  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845  df-chn 32980  df-dim 33627  df-fldext 33670  df-extdg 33671

This theorem is referenced by: (None)