Theorem fldext2chn 33719

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 6920	. . . . . . 7 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
3	2	breq2d 5178	. . . . . 6 ⊢ (𝑑 = ∅ → (0 < (♯‘𝑑) ↔ 0 < (♯‘∅)))
4		fveq2 6920	. . . . . . . 8 ⊢ (𝑑 = ∅ → (lastS‘𝑑) = (lastS‘∅))
5		fveq1 6919	. . . . . . . 8 ⊢ (𝑑 = ∅ → (𝑑‘0) = (∅‘0))
6	4, 5	breq12d 5179	. . . . . . 7 ⊢ (𝑑 = ∅ → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘∅)/FldExt(∅‘0)))
7	4, 5	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑑 = ∅ → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘∅)[:](∅‘0)))
8	7	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑑 = ∅ → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))
9	8	rexbidv 3185	. . . . . . 7 ⊢ (𝑑 = ∅ → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘∅)[:](∅‘0)) = (2↑𝑛)))
10	6, 9	anbi12d 631	. . . . . 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 6920	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (♯‘𝑑) = (♯‘𝑐))
13	12	breq2d 5178	. . . . . 6 ⊢ (𝑑 = 𝑐 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑐)))
14		fveq2 6920	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (lastS‘𝑑) = (lastS‘𝑐))
15		fveq1 6919	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (𝑑‘0) = (𝑐‘0))
16	14, 15	breq12d 5179	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘𝑐)/FldExt(𝑐‘0)))
17	14, 15	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘𝑐)[:](𝑐‘0)))
18	17	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
19	18	rexbidv 3185	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛)))
20	16, 19	anbi12d 631	. . . . . 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 6920	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (♯‘𝑑) = (♯‘(𝑐 ++ ⟨“𝑔”⟩)))
23	22	breq2d 5178	. . . . . 6 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (0 < (♯‘𝑑) ↔ 0 < (♯‘(𝑐 ++ ⟨“𝑔”⟩))))
24		fveq2 6920	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (lastS‘𝑑) = (lastS‘(𝑐 ++ ⟨“𝑔”⟩)))
25		fveq1 6919	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (𝑑‘0) = ((𝑐 ++ ⟨“𝑔”⟩)‘0))
26	24, 25	breq12d 5179	. . . . . . 7 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘(𝑐 ++ ⟨“𝑔”⟩))/FldExt((𝑐 ++ ⟨“𝑔”⟩)‘0)))
27	24, 25	oveq12d 7466	. . . . . . . . . 10 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)))
28	27	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛)))
29	28	rexbidv 3185	. . . . . . . 8 ⊢ (𝑑 = (𝑐 ++ ⟨“𝑔”⟩) → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛)))
30		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
31	30	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑛) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚)))
32	31	cbvrexvw 3244	. . . . . . . 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 631	. . . . . 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 6920	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (♯‘𝑑) = (♯‘𝑇))
37	36	breq2d 5178	. . . . . 6 ⊢ (𝑑 = 𝑇 → (0 < (♯‘𝑑) ↔ 0 < (♯‘𝑇)))
38		fveq2 6920	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (lastS‘𝑑) = (lastS‘𝑇))
39		fveq1 6919	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (𝑑‘0) = (𝑇‘0))
40	38, 39	breq12d 5179	. . . . . . 7 ⊢ (𝑑 = 𝑇 → ((lastS‘𝑑)/FldExt(𝑑‘0) ↔ (lastS‘𝑇)/FldExt(𝑇‘0)))
41	38, 39	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑑 = 𝑇 → ((lastS‘𝑑)[:](𝑑‘0)) = ((lastS‘𝑇)[:](𝑇‘0)))
42	41	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑑 = 𝑇 → (((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
43	42	rexbidv 3185	. . . . . . 7 ⊢ (𝑑 = 𝑇 → (∃𝑛 ∈ ℕ0 ((lastS‘𝑑)[:](𝑑‘0)) = (2↑𝑛) ↔ ∃𝑛 ∈ ℕ0 ((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛)))
44	40, 43	anbi12d 631	. . . . . 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 11292	. . . . . . . . 9 ⊢ 0 ∈ ℝ
48	47	ltnri 11399	. . . . . . . 8 ⊢ ¬ 0 < 0
49	48	a1i 11	. . . . . . 7 ⊢ (𝜑 → ¬ 0 < 0)
50		hash0 14416	. . . . . . . 8 ⊢ (♯‘∅) = 0
51	50	breq2i 5174	. . . . . . 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 785	. . . . . . . . . 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 33672	. . . . . . . . . 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 785	. . . . . . . . . . 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 32980	. . . . . . . . . 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 14682	. . . . . . . . . 10 ⊢ ((𝑐 ∈ Word Field ∧ 𝑔 ∈ Field) → (lastS‘(𝑐 ++ ⟨“𝑔”⟩)) = 𝑔)
60	58, 54, 59	syl2an2r 684	. . . . . . . . 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 7463	. . . . . . . . . . . 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 14971	. . . . . . . . . . . 12 ⊢ ⟨“𝑔”⟩ = (∅ ++ ⟨“𝑔”⟩)
64	62, 63	eqtr4di 2798	. . . . . . . . . . 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 6922	. . . . . . . . . 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 14658	. . . . . . . . . . 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 2780	. . . . . . . . 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 5198	. . . . . . . 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 7456	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (2↑𝑚) = (2↑0))
71		2cn 12368	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
72		exp0 14116	. . . . . . . . . . . 12 ⊢ (2 ∈ ℂ → (2↑0) = 1)
73	71, 72	ax-mp 5	. . . . . . . . . . 11 ⊢ (2↑0) = 1
74	70, 73	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (2↑𝑚) = 1)
75	74	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑚 = 0 → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = 1))
76		0nn0 12568	. . . . . . . . . 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 7466	. . . . . . . . . 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 33673	. . . . . . . . . . 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 2780	. . . . . . . . 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 3638	. . . . . . . 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 787	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 2951	. . . . . . . . . . . . . 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 877	. . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . 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 14616	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ Word Field ∧ 𝑐 ≠ ∅) → (lastS‘𝑐) ∈ Field)
90	88, 85, 89	syl2anc 583	. . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . 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 5171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑒/FldExt𝑓 ↔ 𝑔/FldExt(lastS‘𝑐)))
93		oveq12 7457	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → (𝑒[:]𝑓) = (𝑔[:](lastS‘𝑐)))
94	93	eqeq1d 2742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = 𝑔 ∧ 𝑓 = (lastS‘𝑐)) → ((𝑒[:]𝑓) = 2 ↔ (𝑔[:](lastS‘𝑐)) = 2))
95	92, 94	anbi12d 631	. . . . . . . . . . . . . . . 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 5553	. . . . . . . . . . . . . 14 ⊢ (((lastS‘𝑐) ∈ Field ∧ 𝑔 ∈ Field) → ((lastS‘𝑐) < 𝑔 ↔ (𝑔/FldExt(lastS‘𝑐) ∧ (𝑔[:](lastS‘𝑐)) = 2)))
99	90, 91, 98	syl2anc 583	. . . . . . . . . . . . 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 14437	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ( < ChainField) ∧ 𝑐 ≠ ∅) → 0 < (♯‘𝑐))
103	57, 102	sylan 579	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑜 → (2↑𝑛) = (2↑𝑜))
108	107	eqeq2d 2751	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑜 → (((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑛) ↔ ((lastS‘𝑐)[:](𝑐‘0)) = (2↑𝑜)))
109	108	cbvrexvw 3244	. . . . . . . . . . . 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 3168	. . . . . . . . . 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 33671	. . . . . . . . . 10 ⊢ ((𝑔/FldExt(lastS‘𝑐) ∧ (lastS‘𝑐)/FldExt(𝑐‘0)) → 𝑔/FldExt(𝑐‘0))
114	111, 112, 113	syl2anc 583	. . . . . . . . 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 583	. . . . . . . . . 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 3168	. . . . . . . . 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 14651	. . . . . . . . . . 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 725	. . . . . . . . . . 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 14631	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Word Field ∧ ⟨“𝑔”⟩ ∈ Word Field ∧ 0 < (♯‘𝑐)) → ((𝑐 ++ ⟨“𝑔”⟩)‘0) = (𝑐‘0))
120	88, 117, 118, 119	syl3anc 1371	. . . . . . . . . 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 3168	. . . . . . . . 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 5198	. . . . . . . 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 7456	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑜 + 1) → (2↑𝑚) = (2↑(𝑜 + 1)))
124	123	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑚 = (𝑜 + 1) → (((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑𝑚) ↔ ((lastS‘(𝑐 ++ ⟨“𝑔”⟩))[:]((𝑐 ++ ⟨“𝑔”⟩)‘0)) = (2↑(𝑜 + 1))))
125		simplr 768	. . . . . . . . . . 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 12569	. . . . . . . . . . . 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 12617	. . . . . . . . . 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 7466	. . . . . . . . . . 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 725	. . . . . . . . . . . 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 33674	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 12371	. . . . . . . . . . . . 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 14196	. . . . . . . . . . . . 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 11311	. . . . . . . . . . . 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 7466	. . . . . . . . . . . . 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 12367	. . . . . . . . . . . . . . 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 14213	. . . . . . . . . . . . . 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 13333	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (2↑𝑜) ∈ ℝ) → (2 ·e (2↑𝑜)) = (2 · (2↑𝑜)))
143	140, 141, 142	syl2anc 583	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 14197	. . . . . . . . . . . 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 2790	. . . . . . . . . . 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 2784	. . . . . . . . . 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 3638	. . . . . . . . 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 3168	. . . . . . . 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 3035	. . . . . 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 32983	. . . 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 7466	. . . 4 ⊢ (𝜑 → ((lastS‘𝑇)[:](𝑇‘0)) = (𝐹[:]𝑄))
159	158	eqeq1d 2742	. . 3 ⊢ (𝜑 → (((lastS‘𝑇)[:](𝑇‘0)) = (2↑𝑛) ↔ (𝐹[:]𝑄) = (2↑𝑛)))
160	159	rexbidv 3185	. 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 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  ∅c0 4352   class class class wbr 5166  {copab 5228  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189   < clt 11324  2c2 12348  ℕ₀cn0 12553   ·_e cxmu 13174  ↑cexp 14112  ♯chash 14379  Word cword 14562  lastSclsw 14610   ++ cconcat 14618  ⟨“cs1 14643  Fieldcfield 20752  Chaincchn 32977  /_FldExtcfldext 33651  [:]cextdg 33654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-rpss 7758  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-r1 9833  df-rank 9834  df-dju 9970  df-card 10008  df-acn 10011  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-rp 13058  df-xmul 13177  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ocomp 17332  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-mri 17646  df-acs 17647  df-proset 18365  df-drs 18366  df-poset 18383  df-ipo 18598  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-nzr 20539  df-subrng 20572  df-subrg 20597  df-drng 20753  df-field 20754  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lmhm 21044  df-lbs 21097  df-lvec 21125  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-rsp 21242  df-dsmm 21775  df-frlm 21790  df-uvc 21826  df-lindf 21849  df-linds 21850  df-chn 32978  df-dim 33612  df-fldext 33655  df-extdg 33656

This theorem is referenced by: (None)