chnub - Mathbox for Thierry Arnoux

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Theorem chnub 32938

Description: In a chain, the last element is an upper bound. (Contributed by Thierry Arnoux, 19-Jun-2025.)

**Hypotheses**
Ref	Expression
chnub.1	⊢ (𝜑 → < Po 𝐴)
chnub.2	⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
chnub.3	⊢ (𝜑 → 𝐼 ∈ (0..^((♯‘𝐶) − 1)))

**Assertion**
Ref	Expression
chnub	⊢ (𝜑 → (𝐶‘𝐼) < (lastS‘𝐶))

Proof of Theorem chnub Dummy variables 𝑐 𝑑 𝑥 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . 3 ⊢ (𝑖 = 𝐼 → (𝐶‘𝑖) = (𝐶‘𝐼))
2	1	breq1d 5117	. 2 ⊢ (𝑖 = 𝐼 → ((𝐶‘𝑖) < (lastS‘𝐶) ↔ (𝐶‘𝐼) < (lastS‘𝐶)))
3		fveq2 6858	. . . . . 6 ⊢ (𝑐 = ∅ → (♯‘𝑐) = (♯‘∅))
4	3	oveq1d 7402	. . . . 5 ⊢ (𝑐 = ∅ → ((♯‘𝑐) − 1) = ((♯‘∅) − 1))
5	4	oveq2d 7403	. . . 4 ⊢ (𝑐 = ∅ → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘∅) − 1)))
6		fveq1 6857	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐‘𝑖) = (∅‘𝑖))
7		fveq2 6858	. . . . 5 ⊢ (𝑐 = ∅ → (lastS‘𝑐) = (lastS‘∅))
8	6, 7	breq12d 5120	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (∅‘𝑖) < (lastS‘∅)))
9	5, 8	raleqbidv 3319	. . 3 ⊢ (𝑐 = ∅ → (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅)))
10		fveq2 6858	. . . . . 6 ⊢ (𝑐 = 𝑑 → (♯‘𝑐) = (♯‘𝑑))
11	10	oveq1d 7402	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) − 1) = ((♯‘𝑑) − 1))
12	11	oveq2d 7403	. . . 4 ⊢ (𝑐 = 𝑑 → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘𝑑) − 1)))
13		fveq1 6857	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑖) = (𝑑‘𝑖))
14		fveq2 6858	. . . . 5 ⊢ (𝑐 = 𝑑 → (lastS‘𝑐) = (lastS‘𝑑))
15	13, 14	breq12d 5120	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (𝑑‘𝑖) < (lastS‘𝑑)))
16	12, 15	raleqbidv 3319	. . 3 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)))
17		fveq2 6858	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
18	17	breq1d 5117	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (𝑐‘𝑗) < (lastS‘𝑐)))
19	18	cbvralvw 3215	. . . 4 ⊢ (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑗) < (lastS‘𝑐))
20		fveq2 6858	. . . . . . 7 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (♯‘𝑐) = (♯‘(𝑑 ++ ⟨“𝑥”⟩)))
21	20	oveq1d 7402	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → ((♯‘𝑐) − 1) = ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))
22	21	oveq2d 7403	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)))
23		fveq1 6857	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (𝑐‘𝑗) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
24		fveq2 6858	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (lastS‘𝑐) = (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
25	23, 24	breq12d 5120	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → ((𝑐‘𝑗) < (lastS‘𝑐) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
26	22, 25	raleqbidv 3319	. . . 4 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (∀𝑗 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑗) < (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
27	19, 26	bitrid 283	. . 3 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
28		fveq2 6858	. . . . . 6 ⊢ (𝑐 = 𝐶 → (♯‘𝑐) = (♯‘𝐶))
29	28	oveq1d 7402	. . . . 5 ⊢ (𝑐 = 𝐶 → ((♯‘𝑐) − 1) = ((♯‘𝐶) − 1))
30	29	oveq2d 7403	. . . 4 ⊢ (𝑐 = 𝐶 → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘𝐶) − 1)))
31		fveq1 6857	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
32		fveq2 6858	. . . . 5 ⊢ (𝑐 = 𝐶 → (lastS‘𝑐) = (lastS‘𝐶))
33	31, 32	breq12d 5120	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (𝐶‘𝑖) < (lastS‘𝐶)))
34	30, 33	raleqbidv 3319	. . 3 ⊢ (𝑐 = 𝐶 → (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^((♯‘𝐶) − 1))(𝐶‘𝑖) < (lastS‘𝐶)))
35		chnub.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
36		ral0 4476	. . . . 5 ⊢ ∀𝑖 ∈ ∅ (∅‘𝑖) < (lastS‘∅)
37		hash0 14332	. . . . . . . . 9 ⊢ (♯‘∅) = 0
38	37	oveq1i 7397	. . . . . . . 8 ⊢ ((♯‘∅) − 1) = (0 − 1)
39		df-neg 11408	. . . . . . . . 9 ⊢ -1 = (0 − 1)
40		neg1rr 12172	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
41		0re 11176	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
42		neg1lt0 12174	. . . . . . . . . 10 ⊢ -1 < 0
43	40, 41, 42	ltleii 11297	. . . . . . . . 9 ⊢ -1 ≤ 0
44	39, 43	eqbrtrri 5130	. . . . . . . 8 ⊢ (0 − 1) ≤ 0
45	38, 44	eqbrtri 5128	. . . . . . 7 ⊢ ((♯‘∅) − 1) ≤ 0
46		0z 12540	. . . . . . . 8 ⊢ 0 ∈ ℤ
47	37, 46	eqeltri 2824	. . . . . . . . 9 ⊢ (♯‘∅) ∈ ℤ
48		1z 12563	. . . . . . . . 9 ⊢ 1 ∈ ℤ
49		zsubcl 12575	. . . . . . . . 9 ⊢ (((♯‘∅) ∈ ℤ ∧ 1 ∈ ℤ) → ((♯‘∅) − 1) ∈ ℤ)
50	47, 48, 49	mp2an 692	. . . . . . . 8 ⊢ ((♯‘∅) − 1) ∈ ℤ
51		fzon 13641	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ ((♯‘∅) − 1) ∈ ℤ) → (((♯‘∅) − 1) ≤ 0 ↔ (0..^((♯‘∅) − 1)) = ∅))
52	46, 50, 51	mp2an 692	. . . . . . 7 ⊢ (((♯‘∅) − 1) ≤ 0 ↔ (0..^((♯‘∅) − 1)) = ∅)
53	45, 52	mpbi 230	. . . . . 6 ⊢ (0..^((♯‘∅) − 1)) = ∅
54	53	raleqi 3297	. . . . 5 ⊢ (∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅) ↔ ∀𝑖 ∈ ∅ (∅‘𝑖) < (lastS‘∅))
55	36, 54	mpbir 231	. . . 4 ⊢ ∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅)
56	55	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅))
57		simp-6r 787	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → 𝑑 ∈ ( < Chain𝐴))
58	57	chnwrd 32933	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → 𝑑 ∈ Word 𝐴)
59		ccatws1len 14585	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
61		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → 𝑑 = ∅)
62	61	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (♯‘𝑑) = (♯‘∅))
63	62, 37	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (♯‘𝑑) = 0)
64	63	oveq1d 7402	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → ((♯‘𝑑) + 1) = (0 + 1))
65		0p1e1 12303	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
66	65	a1i 11	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (0 + 1) = 1)
67	60, 64, 66	3eqtrd 2768	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = 1)
68	67	oveq1d 7402	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1) = (1 − 1))
69		1m1e0 12258	. . . . . . . . 9 ⊢ (1 − 1) = 0
70	68, 69	eqtrdi 2780	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1) = 0)
71	70	oveq2d 7403	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)) = (0..^0))
72		fzo0 13644	. . . . . . 7 ⊢ (0..^0) = ∅
73	71, 72	eqtrdi 2780	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)) = ∅)
74		simplr 768	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)))
75	74	ne0d 4305	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)) ≠ ∅)
76	73, 75	pm2.21ddne 3009	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
77		chnub.1	. . . . . . . 8 ⊢ (𝜑 → < Po 𝐴)
78	77	ad7antr 738	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → < Po 𝐴)
79		simp-6r 787	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑑 ∈ ( < Chain𝐴))
80	79	chnwrd 32933	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑑 ∈ Word 𝐴)
81	80	adantr 480	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑑 ∈ Word 𝐴)
82		simp-5r 785	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑥 ∈ 𝐴)
83	82	adantr 480	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑥 ∈ 𝐴)
84		ccatws1cl 14581	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑑 ++ ⟨“𝑥”⟩) ∈ Word 𝐴)
85	81, 83, 84	syl2anc 584	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (𝑑 ++ ⟨“𝑥”⟩) ∈ Word 𝐴)
86		lencl 14498	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘𝑑) ∈ ℕ₀)
87	80, 86	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℕ₀)
88	87	nn0zd 12555	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℤ)
89		1zzd 12564	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 1 ∈ ℤ)
90	88, 89	zsubcld 12643	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) − 1) ∈ ℤ)
91	88	peano2zd 12641	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) + 1) ∈ ℤ)
92	90	zred 12638	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) − 1) ∈ ℝ)
93	91	zred 12638	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) + 1) ∈ ℝ)
94		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑑 ≠ ∅)
95		hasheq0 14328	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ Word 𝐴 → ((♯‘𝑑) = 0 ↔ 𝑑 = ∅))
96	95	necon3bid 2969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ Word 𝐴 → ((♯‘𝑑) ≠ 0 ↔ 𝑑 ≠ ∅))
97	96	biimpar 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ≠ 0)
98	80, 94, 97	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ≠ 0)
99		elnnne0 12456	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑑) ∈ ℕ ↔ ((♯‘𝑑) ∈ ℕ₀ ∧ (♯‘𝑑) ≠ 0))
100	87, 98, 99	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℕ)
101	100	nnred 12201	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℝ)
102	101	ltm1d 12115	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) − 1) < (♯‘𝑑))
103	101	ltp1d 12113	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) < ((♯‘𝑑) + 1))
104	92, 101, 93, 102, 103	lttrd 11335	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) − 1) < ((♯‘𝑑) + 1))
105	92, 93, 104	ltled 11322	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) − 1) ≤ ((♯‘𝑑) + 1))
106		eluz2 12799	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑑) + 1) ∈ (ℤ_≥‘((♯‘𝑑) − 1)) ↔ (((♯‘𝑑) − 1) ∈ ℤ ∧ ((♯‘𝑑) + 1) ∈ ℤ ∧ ((♯‘𝑑) − 1) ≤ ((♯‘𝑑) + 1)))
107	90, 91, 105, 106	syl3anbrc 1344	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘𝑑) + 1) ∈ (ℤ_≥‘((♯‘𝑑) − 1)))
108		fzoss2 13648	. . . . . . . . . . . 12 ⊢ (((♯‘𝑑) + 1) ∈ (ℤ_≥‘((♯‘𝑑) − 1)) → (0..^((♯‘𝑑) − 1)) ⊆ (0..^((♯‘𝑑) + 1)))
109	107, 108	syl 17	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (0..^((♯‘𝑑) − 1)) ⊆ (0..^((♯‘𝑑) + 1)))
110	109	sselda 3946	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑗 ∈ (0..^((♯‘𝑑) + 1)))
111	81, 59	syl 17	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
112	111	oveq2d 7403	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))) = (0..^((♯‘𝑑) + 1)))
113	110, 112	eleqtrrd 2831	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))))
114		wrdsymbcl 14492	. . . . . . . . 9 ⊢ (((𝑑 ++ ⟨“𝑥”⟩) ∈ Word 𝐴 ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∈ 𝐴)
115	85, 113, 114	syl2anc 584	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∈ 𝐴)
116		simplr 768	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑑 ≠ ∅)
117		lswcl 14533	. . . . . . . . 9 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑑 ≠ ∅) → (lastS‘𝑑) ∈ 𝐴)
118	81, 116, 117	syl2anc 584	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (lastS‘𝑑) ∈ 𝐴)
119		lswccats1 14599	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
120	80, 82, 119	syl2anc 584	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
121	120	adantr 480	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
122	121, 83	eqeltrd 2828	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) ∈ 𝐴)
123	115, 118, 122	3jca 1128	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∈ 𝐴 ∧ (lastS‘𝑑) ∈ 𝐴 ∧ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) ∈ 𝐴))
124		simplr 768	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)))
125	59	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ Word 𝐴 → ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1) = (((♯‘𝑑) + 1) − 1))
126	80, 125	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1) = (((♯‘𝑑) + 1) − 1))
127	100	nncnd 12202	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℂ)
128		1cnd 11169	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 1 ∈ ℂ)
129	127, 128	pncand 11534	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (((♯‘𝑑) + 1) − 1) = (♯‘𝑑))
130	126, 129	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1) = (♯‘𝑑))
131	130	oveq2d 7403	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)) = (0..^(♯‘𝑑)))
132	124, 131	eleqtrd 2830	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑗 ∈ (0..^(♯‘𝑑)))
133	132	adantr 480	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑗 ∈ (0..^(♯‘𝑑)))
134		ccats1val1 14591	. . . . . . . . 9 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑗 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = (𝑑‘𝑗))
135	81, 133, 134	syl2anc 584	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = (𝑑‘𝑗))
136		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗))
137	136	breq1d 5117	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖) < (lastS‘𝑑) ↔ (𝑑‘𝑗) < (lastS‘𝑑)))
138		simp-4r 783	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑))
139		simpr 484	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → 𝑗 ∈ (0..^((♯‘𝑑) − 1)))
140	137, 138, 139	rspcdva 3589	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (𝑑‘𝑗) < (lastS‘𝑑))
141	135, 140	eqbrtrd 5129	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘𝑑))
142		simp-4r 783	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥))
143	94	neneqd 2930	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ¬ 𝑑 = ∅)
144	142, 143	orcnd 878	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (lastS‘𝑑) < 𝑥)
145	144	adantr 480	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (lastS‘𝑑) < 𝑥)
146	145, 121	breqtrrd 5135	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → (lastS‘𝑑) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
147		potr 5559	. . . . . . . 8 ⊢ (( < Po 𝐴 ∧ (((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∈ 𝐴 ∧ (lastS‘𝑑) ∈ 𝐴 ∧ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) ∈ 𝐴)) → ((((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘𝑑) ∧ (lastS‘𝑑) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
148	147	imp 406	. . . . . . 7 ⊢ ((( < Po 𝐴 ∧ (((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∈ 𝐴 ∧ (lastS‘𝑑) ∈ 𝐴 ∧ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) ∈ 𝐴)) ∧ (((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘𝑑) ∧ (lastS‘𝑑) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
149	78, 123, 141, 146, 148	syl22anc 838	. . . . . 6 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ∈ (0..^((♯‘𝑑) − 1))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
150	144	adantr 480	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → (lastS‘𝑑) < 𝑥)
151	80	adantr 480	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → 𝑑 ∈ Word 𝐴)
152		simp-6r 787	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → 𝑥 ∈ 𝐴)
153	152	s1cld 14568	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → ⟨“𝑥”⟩ ∈ Word 𝐴)
154	100	adantr 480	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → (♯‘𝑑) ∈ ℕ)
155		fzo0end 13719	. . . . . . . . . 10 ⊢ ((♯‘𝑑) ∈ ℕ → ((♯‘𝑑) − 1) ∈ (0..^(♯‘𝑑)))
156	154, 155	syl 17	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → ((♯‘𝑑) − 1) ∈ (0..^(♯‘𝑑)))
157		ccatval1 14542	. . . . . . . . 9 ⊢ ((𝑑 ∈ Word 𝐴 ∧ ⟨“𝑥”⟩ ∈ Word 𝐴 ∧ ((♯‘𝑑) − 1) ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘((♯‘𝑑) − 1)) = (𝑑‘((♯‘𝑑) − 1)))
158	151, 153, 156, 157	syl3anc 1373	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → ((𝑑 ++ ⟨“𝑥”⟩)‘((♯‘𝑑) − 1)) = (𝑑‘((♯‘𝑑) − 1)))
159		simpr 484	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → 𝑗 = ((♯‘𝑑) − 1))
160	159	fveq2d 6862	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = ((𝑑 ++ ⟨“𝑥”⟩)‘((♯‘𝑑) − 1)))
161		lsw 14529	. . . . . . . . 9 ⊢ (𝑑 ∈ Word 𝐴 → (lastS‘𝑑) = (𝑑‘((♯‘𝑑) − 1)))
162	151, 161	syl 17	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → (lastS‘𝑑) = (𝑑‘((♯‘𝑑) − 1)))
163	158, 160, 162	3eqtr4d 2774	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = (lastS‘𝑑))
164	120	adantr 480	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
165	150, 163, 164	3brtr4d 5139	. . . . . 6 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = ((♯‘𝑑) − 1)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
166	65	fveq2i 6861	. . . . . . . . . . 11 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
167		nnuz 12836	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
168	166, 167	eqtr4i 2755	. . . . . . . . . 10 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
169	100, 168	eleqtrrdi 2839	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ (ℤ_≥‘(0 + 1)))
170		fzosplitsnm1 13701	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (♯‘𝑑) ∈ (ℤ_≥‘(0 + 1))) → (0..^(♯‘𝑑)) = ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}))
171	46, 169, 170	sylancr 587	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (0..^(♯‘𝑑)) = ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}))
172	132, 171	eleqtrd 2830	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}))
173		elunsn 4647	. . . . . . . 8 ⊢ (𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}) → (𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}) ↔ (𝑗 ∈ (0..^((♯‘𝑑) − 1)) ∨ 𝑗 = ((♯‘𝑑) − 1))))
174	173	ibi 267	. . . . . . 7 ⊢ (𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}) → (𝑗 ∈ (0..^((♯‘𝑑) − 1)) ∨ 𝑗 = ((♯‘𝑑) − 1)))
175	172, 174	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (𝑗 ∈ (0..^((♯‘𝑑) − 1)) ∨ 𝑗 = ((♯‘𝑑) − 1)))
176	149, 165, 175	mpjaodan 960	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
177	76, 176	pm2.61dane 3012	. . . 4 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
178	177	ralrimiva 3125	. . 3 ⊢ (((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) → ∀𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
179	9, 16, 27, 34, 35, 56, 178	chnind 32937	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^((♯‘𝐶) − 1))(𝐶‘𝑖) < (lastS‘𝐶))
180		chnub.3	. 2 ⊢ (𝜑 → 𝐼 ∈ (0..^((♯‘𝐶) − 1)))
181	2, 179, 180	rspcdva 3589	1 ⊢ (𝜑 → (𝐶‘𝐼) < (lastS‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∪ cun 3912 ⊆ wss 3914 ∅c0 4296 {csn 4589 class class class wbr 5107 Po wpo 5544 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 + caddc 11071 ≤ cle 11209 − cmin 11405 -cneg 11406 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ..^cfzo 13615 ♯chash 14295 Word cword 14478 lastSclsw 14527 ++ cconcat 14535 ⟨“cs1 14560 Chaincchn 32930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-hash 14296 df-word 14479 df-lsw 14528 df-concat 14536 df-s1 14561 df-substr 14606 df-pfx 14636 df-chn 32931

This theorem is referenced by: chnlt 32939

W3C validator