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Theorem chnub 18679

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 ∈ ( < Chain 𝐴))
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 6883	. . 3 ⊢ (𝑖 = 𝐼 → (𝐶‘𝑖) = (𝐶‘𝐼))
2	1	breq1d 5120	. 2 ⊢ (𝑖 = 𝐼 → ((𝐶‘𝑖) < (lastS‘𝐶) ↔ (𝐶‘𝐼) < (lastS‘𝐶)))
3		fveq2 6883	. . . . . 6 ⊢ (𝑐 = ∅ → (♯‘𝑐) = (♯‘∅))
4	3	oveq1d 7427	. . . . 5 ⊢ (𝑐 = ∅ → ((♯‘𝑐) − 1) = ((♯‘∅) − 1))
5	4	oveq2d 7428	. . . 4 ⊢ (𝑐 = ∅ → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘∅) − 1)))
6		fveq1 6882	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐‘𝑖) = (∅‘𝑖))
7		fveq2 6883	. . . . 5 ⊢ (𝑐 = ∅ → (lastS‘𝑐) = (lastS‘∅))
8	6, 7	breq12d 5123	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (∅‘𝑖) < (lastS‘∅)))
9	5, 8	raleqbidv 3338	. . 3 ⊢ (𝑐 = ∅ → (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅)))
10		fveq2 6883	. . . . . 6 ⊢ (𝑐 = 𝑑 → (♯‘𝑐) = (♯‘𝑑))
11	10	oveq1d 7427	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) − 1) = ((♯‘𝑑) − 1))
12	11	oveq2d 7428	. . . 4 ⊢ (𝑐 = 𝑑 → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘𝑑) − 1)))
13		fveq1 6882	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑖) = (𝑑‘𝑖))
14		fveq2 6883	. . . . 5 ⊢ (𝑐 = 𝑑 → (lastS‘𝑐) = (lastS‘𝑑))
15	13, 14	breq12d 5123	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (𝑑‘𝑖) < (lastS‘𝑑)))
16	12, 15	raleqbidv 3338	. . 3 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)))
17		fveq2 6883	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
18	17	breq1d 5120	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (𝑐‘𝑗) < (lastS‘𝑐)))
19	18	cbvralvw 3243	. . . 4 ⊢ (∀𝑖 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑖) < (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^((♯‘𝑐) − 1))(𝑐‘𝑗) < (lastS‘𝑐))
20		fveq2 6883	. . . . . . 7 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (♯‘𝑐) = (♯‘(𝑑 ++ ⟨“𝑥”⟩)))
21	20	oveq1d 7427	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → ((♯‘𝑐) − 1) = ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))
22	21	oveq2d 7428	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1)))
23		fveq1 6882	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (𝑐‘𝑗) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
24		fveq2 6883	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (lastS‘𝑐) = (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
25	23, 24	breq12d 5123	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → ((𝑐‘𝑗) < (lastS‘𝑐) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) < (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
26	22, 25	raleqbidv 3338	. . . 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 286	. . 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 6883	. . . . . 6 ⊢ (𝑐 = 𝐶 → (♯‘𝑐) = (♯‘𝐶))
29	28	oveq1d 7427	. . . . 5 ⊢ (𝑐 = 𝐶 → ((♯‘𝑐) − 1) = ((♯‘𝐶) − 1))
30	29	oveq2d 7428	. . . 4 ⊢ (𝑐 = 𝐶 → (0..^((♯‘𝑐) − 1)) = (0..^((♯‘𝐶) − 1)))
31		fveq1 6882	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
32		fveq2 6883	. . . . 5 ⊢ (𝑐 = 𝐶 → (lastS‘𝑐) = (lastS‘𝐶))
33	31, 32	breq12d 5123	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑖) < (lastS‘𝑐) ↔ (𝐶‘𝑖) < (lastS‘𝐶)))
34	30, 33	raleqbidv 3338	. . 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 ∈ ( < Chain 𝐴))
36		ral0 4460	. . . . 5 ⊢ ∀𝑖 ∈ ∅ (∅‘𝑖) < (lastS‘∅)
37		hash0 14405	. . . . . . . . 9 ⊢ (♯‘∅) = 0
38	37	oveq1i 7422	. . . . . . . 8 ⊢ ((♯‘∅) − 1) = (0 − 1)
39		df-neg 11445	. . . . . . . . 9 ⊢ -1 = (0 − 1)
40		neg1rr 12205	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
41		0re 11211	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
42		neg1lt0 12207	. . . . . . . . . 10 ⊢ -1 < 0
43	40, 41, 42	ltleii 11334	. . . . . . . . 9 ⊢ -1 ≤ 0
44	39, 43	eqbrtrri 5135	. . . . . . . 8 ⊢ (0 − 1) ≤ 0
45	38, 44	eqbrtri 5133	. . . . . . 7 ⊢ ((♯‘∅) − 1) ≤ 0
46		0z 12603	. . . . . . . 8 ⊢ 0 ∈ ℤ
47	37, 46	eqeltri 2859	. . . . . . . . 9 ⊢ (♯‘∅) ∈ ℤ
48		1z 12625	. . . . . . . . 9 ⊢ 1 ∈ ℤ
49		zsubcl 12637	. . . . . . . . 9 ⊢ (((♯‘∅) ∈ ℤ ∧ 1 ∈ ℤ) → ((♯‘∅) − 1) ∈ ℤ)
50	47, 48, 49	mp2an 704	. . . . . . . 8 ⊢ ((♯‘∅) − 1) ∈ ℤ
51		fzon 13711	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ ((♯‘∅) − 1) ∈ ℤ) → (((♯‘∅) − 1) ≤ 0 ↔ (0..^((♯‘∅) − 1)) = ∅))
52	46, 50, 51	mp2an 704	. . . . . . 7 ⊢ (((♯‘∅) − 1) ≤ 0 ↔ (0..^((♯‘∅) − 1)) = ∅)
53	45, 52	mpbi 233	. . . . . 6 ⊢ (0..^((♯‘∅) − 1)) = ∅
54	53	raleqi 3321	. . . . 5 ⊢ (∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅) ↔ ∀𝑖 ∈ ∅ (∅‘𝑖) < (lastS‘∅))
55	36, 54	mpbir 234	. . . 4 ⊢ ∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅)
56	55	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^((♯‘∅) − 1))(∅‘𝑖) < (lastS‘∅))
57		simp-6r 799	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → 𝑑 ∈ ( < Chain 𝐴))
58	57	chnwrd 18665	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14660	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
60	58, 59	syl 18	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 489	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 = ∅) → 𝑑 = ∅)
62	61	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2814	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 7427	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12362	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
66	65	a1i 11	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2802	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 7427	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12314	. . . . . . . . 9 ⊢ (1 − 1) = 0
70	68, 69	eqtrdi 2814	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 7428	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 13714	. . . . . . 7 ⊢ (0..^0) = ∅
73	71, 72	eqtrdi 2814	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 780	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 4296	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 3042	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 750	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 799	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑑 ∈ ( < Chain 𝐴))
80	79	chnwrd 18665	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 797	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14656	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑑 ++ ⟨“𝑥”⟩) ∈ Word 𝐴)
85	81, 83, 84	syl2anc 595	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14572	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘𝑑) ∈ ℕ₀)
87	80, 86	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12617	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℤ)
89		1zzd 12626	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12706	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12704	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12701	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12701	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → 𝑑 ≠ ∅)
95		hasheq0 14401	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ Word 𝐴 → ((♯‘𝑑) = 0 ↔ 𝑑 = ∅))
96	95	necon3bid 3002	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ Word 𝐴 → ((♯‘𝑑) ≠ 0 ↔ 𝑑 ≠ ∅))
97	96	biimpar 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ≠ 0)
98	80, 94, 97	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ≠ 0)
99		elnnne0 12519	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑑) ∈ ℕ ↔ ((♯‘𝑑) ∈ ℕ₀ ∧ (♯‘𝑑) ≠ 0))
100	87, 98, 99	sylanbrc 594	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12249	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12148	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12146	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 11372	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 11359	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12869	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑑) + 1) ∈ (ℤ_≥‘((♯‘𝑑) − 1)) ↔ (((♯‘𝑑) − 1) ∈ ℤ ∧ ((♯‘𝑑) + 1) ∈ ℤ ∧ ((♯‘𝑑) − 1) ≤ ((♯‘𝑑) + 1)))
107	90, 91, 105, 106	syl3anbrc 1362	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 13718	. . . . . . . . . . . 12 ⊢ (((♯‘𝑑) + 1) ∈ (ℤ_≥‘((♯‘𝑑) − 1)) → (0..^((♯‘𝑑) − 1)) ⊆ (0..^((♯‘𝑑) + 1)))
109	107, 108	syl 18	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 3938	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 18	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 7428	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2866	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14566	. . . . . . . . 9 ⊢ (((𝑑 ++ ⟨“𝑥”⟩) ∈ Word 𝐴 ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∈ 𝐴)
115	85, 113, 114	syl2anc 595	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 780	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14607	. . . . . . . . 9 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑑 ≠ ∅) → (lastS‘𝑑) ∈ 𝐴)
118	81, 116, 117	syl2anc 595	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14674	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
120	80, 82, 119	syl2anc 595	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2863	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 1146	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 780	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 7427	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ Word 𝐴 → ((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1) = (((♯‘𝑑) + 1) − 1))
126	80, 125	syl 18	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 12250	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑑) − 1))(𝑑‘𝑖) < (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^((♯‘(𝑑 ++ ⟨“𝑥”⟩)) − 1))) ∧ 𝑑 ≠ ∅) → (♯‘𝑑) ∈ ℂ)
128		1cnd 11203	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 11571	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2798	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 7428	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2865	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14666	. . . . . . . . 9 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑗 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = (𝑑‘𝑗))
135	81, 133, 134	syl2anc 595	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 6883	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗))
137	136	breq1d 5120	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖) < (lastS‘𝑑) ↔ (𝑑‘𝑗) < (lastS‘𝑑)))
138		simp-4r 795	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 489	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 3583	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 5134	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 795	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2963	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 891	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 5140	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 5584	. . . . . . . 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 411	. . . . . . 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 851	. . . . . 6 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 799	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14643	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 13789	. . . . . . . . . 10 ⊢ ((♯‘𝑑) ∈ ℕ → ((♯‘𝑑) − 1) ∈ (0..^(♯‘𝑑)))
156	154, 155	syl 18	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14616	. . . . . . . . 9 ⊢ ((𝑑 ∈ Word 𝐴 ∧ ⟨“𝑥”⟩ ∈ Word 𝐴 ∧ ((♯‘𝑑) − 1) ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘((♯‘𝑑) − 1)) = (𝑑‘((♯‘𝑑) − 1)))
158	151, 153, 156, 157	syl3anc 1398	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 489	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 6887	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 14603	. . . . . . . . 9 ⊢ (𝑑 ∈ Word 𝐴 → (lastS‘𝑑) = (𝑑‘((♯‘𝑑) − 1)))
162	151, 161	syl 18	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2808	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 485	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 5144	. . . . . 6 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 6886	. . . . . . . . . . 11 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
167		nnuz 12902	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
168	166, 167	eqtr4i 2789	. . . . . . . . . 10 ⊢ (ℤ_≥‘(0 + 1)) = ℕ
169	100, 168	eleqtrrdi 2874	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 13771	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (♯‘𝑑) ∈ (ℤ_≥‘(0 + 1))) → (0..^(♯‘𝑑)) = ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}))
171	46, 169, 170	sylancr 598	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 2865	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 4650	. . . . . . . 8 ⊢ (𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}) → (𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}) ↔ (𝑗 ∈ (0..^((♯‘𝑑) − 1)) ∨ 𝑗 = ((♯‘𝑑) − 1))))
174	173	ibi 270	. . . . . . 7 ⊢ (𝑗 ∈ ((0..^((♯‘𝑑) − 1)) ∪ {((♯‘𝑑) − 1)}) → (𝑗 ∈ (0..^((♯‘𝑑) − 1)) ∨ 𝑗 = ((♯‘𝑑) − 1)))
175	172, 174	syl 18	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 973	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 3045	. . . 4 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 3157	. . 3 ⊢ (((((𝜑 ∧ 𝑑 ∈ ( < Chain 𝐴)) ∧ 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 18678	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^((♯‘𝐶) − 1))(𝐶‘𝑖) < (lastS‘𝐶))
180		chnub.3	. 2 ⊢ (𝜑 → 𝐼 ∈ (0..^((♯‘𝐶) − 1)))
181	2, 179, 180	rspcdva 3583	1 ⊢ (𝜑 → (𝐶‘𝐼) < (lastS‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∪ cun 3904 ⊆ wss 3906 ∅c0 4287 {csn 4590 class class class wbr 5110 Po wpo 5569 ‘cfv 6538 (class class class)co 7412 0cc0 11101 1c1 11102 + caddc 11104 ≤ cle 11245 − cmin 11442 -cneg 11443 ℕcn 12234 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ..^cfzo 13684 ♯chash 14368 Word cword 14552 lastSclsw 14601 ++ cconcat 14609 ⟨“cs1 14635 Chain cchn 18662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-hash 14369 df-word 14553 df-lsw 14602 df-concat 14610 df-s1 14636 df-substr 14681 df-pfx 14711 df-chn 18663

This theorem is referenced by: chnlt 18680

W3C validator