chnerlem1 - Mathbox for Ender Ting

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Theorem chnerlem1 47598

Description: In a chain constructed on an equivalence relation, the last element is equivalent to any. This theorem is a translation of chnub 18673 to equivalence relations. (Contributed by Ender Ting, 29-Jan-2026.)

**Hypotheses**
Ref	Expression
chner.1	⊢ (𝜑 → ∼ Er 𝐴)
chner.2	⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))
chner.3	⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))

**Assertion**
Ref	Expression
chnerlem1	⊢ (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))

Proof of Theorem chnerlem1 Dummy variables 𝑐 𝑑 𝑖 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . 3 ⊢ (𝑖 = 𝐽 → (𝐶‘𝑖) = (𝐶‘𝐽))
2	1	breq1d 5119	. 2 ⊢ (𝑖 = 𝐽 → ((𝐶‘𝑖) ∼ (lastS‘𝐶) ↔ (𝐶‘𝐽) ∼ (lastS‘𝐶)))
3		fveq2 6881	. . . . 5 ⊢ (𝑐 = ∅ → (♯‘𝑐) = (♯‘∅))
4	3	oveq2d 7426	. . . 4 ⊢ (𝑐 = ∅ → (0..^(♯‘𝑐)) = (0..^(♯‘∅)))
5		fveq1 6880	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐‘𝑖) = (∅‘𝑖))
6		fveq2 6881	. . . . 5 ⊢ (𝑐 = ∅ → (lastS‘𝑐) = (lastS‘∅))
7	5, 6	breq12d 5122	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (∅‘𝑖) ∼ (lastS‘∅)))
8	4, 7	raleqbidv 3338	. . 3 ⊢ (𝑐 = ∅ → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅)))
9		fveq2 6881	. . . . 5 ⊢ (𝑐 = 𝑑 → (♯‘𝑐) = (♯‘𝑑))
10	9	oveq2d 7426	. . . 4 ⊢ (𝑐 = 𝑑 → (0..^(♯‘𝑐)) = (0..^(♯‘𝑑)))
11		fveq1 6880	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑖) = (𝑑‘𝑖))
12		fveq2 6881	. . . . 5 ⊢ (𝑐 = 𝑑 → (lastS‘𝑐) = (lastS‘𝑑))
13	11, 12	breq12d 5122	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝑑‘𝑖) ∼ (lastS‘𝑑)))
14	10, 13	raleqbidv 3338	. . 3 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)))
15		fveq2 6881	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
16	15	breq1d 5119	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝑐‘𝑗) ∼ (lastS‘𝑐)))
17	16	cbvralvw 3243	. . . 4 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^(♯‘𝑐))(𝑐‘𝑗) ∼ (lastS‘𝑐))
18		fveq2 6881	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (♯‘𝑐) = (♯‘(𝑑 ++ ⟨“𝑥”⟩)))
19	18	oveq2d 7426	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (0..^(♯‘𝑐)) = (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))))
20		fveq1 6880	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (𝑐‘𝑗) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
21		fveq2 6881	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (lastS‘𝑐) = (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
22	20, 21	breq12d 5122	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → ((𝑐‘𝑗) ∼ (lastS‘𝑐) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
23	19, 22	raleqbidv 3338	. . . 4 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (∀𝑗 ∈ (0..^(♯‘𝑐))(𝑐‘𝑗) ∼ (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
24	17, 23	bitrid 286	. . 3 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
25		fveq2 6881	. . . . 5 ⊢ (𝑐 = 𝐶 → (♯‘𝑐) = (♯‘𝐶))
26	25	oveq2d 7426	. . . 4 ⊢ (𝑐 = 𝐶 → (0..^(♯‘𝑐)) = (0..^(♯‘𝐶)))
27		fveq1 6880	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
28		fveq2 6881	. . . . 5 ⊢ (𝑐 = 𝐶 → (lastS‘𝑐) = (lastS‘𝐶))
29	27, 28	breq12d 5122	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝐶‘𝑖) ∼ (lastS‘𝐶)))
30	26, 29	raleqbidv 3338	. . 3 ⊢ (𝑐 = 𝐶 → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) ∼ (lastS‘𝐶)))
31		chner.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))
32		0nnn 12267	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
33		hash0 14399	. . . . . . . 8 ⊢ (♯‘∅) = 0
34	33	eleq1i 2854	. . . . . . 7 ⊢ ((♯‘∅) ∈ ℕ ↔ 0 ∈ ℕ)
35	32, 34	mtbir 326	. . . . . 6 ⊢ ¬ (♯‘∅) ∈ ℕ
36		fzo0n0 13741	. . . . . 6 ⊢ ((0..^(♯‘∅)) ≠ ∅ ↔ (♯‘∅) ∈ ℕ)
37	35, 36	mtbir 326	. . . . 5 ⊢ ¬ (0..^(♯‘∅)) ≠ ∅
38		nne 2962	. . . . 5 ⊢ (¬ (0..^(♯‘∅)) ≠ ∅ ↔ (0..^(♯‘∅)) = ∅)
39	37, 38	mpbi 233	. . . 4 ⊢ (0..^(♯‘∅)) = ∅
40		rzal 4455	. . . 4 ⊢ ((0..^(♯‘∅)) = ∅ → ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅))
41	39, 40	mp1i 14	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅))
42		chner.1	. . . . . . . 8 ⊢ (𝜑 → ∼ Er 𝐴)
43	42	ad6antr 748	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → ∼ Er 𝐴)
44		simp-5r 797	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑥 ∈ 𝐴)
45	43, 44	erref 8711	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑥 ∼ 𝑥)
46		simp-6r 799	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑑 ∈ ( ∼ Chain 𝐴))
47	46	chnwrd 18659	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑑 ∈ Word 𝐴)
48		simplr 780	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))))
49		ccatws1len 14654	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
50	47, 49	syl 18	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
51		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
52	51, 33	eqtr2di 2815	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ∅ → 0 = (♯‘𝑑))
53	52	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ∅ → (♯‘𝑑) = 0)
54	53	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → (♯‘𝑑) = 0)
55	54	oveq1d 7425	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → ((♯‘𝑑) + 1) = (0 + 1))
56		0p1e1 12356	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
57	55, 56	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → ((♯‘𝑑) + 1) = 1)
58	50, 57	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = 1)
59	58	oveq2d 7426	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))) = (0..^1))
60	48, 59	eleqtrd 2865	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑗 ∈ (0..^1))
61		fzo01 13772	. . . . . . . . . . 11 ⊢ (0..^1) = {0}
62	61	a1i 11	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → (0..^1) = {0})
63	60, 62	eleqtrd 2865	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑗 ∈ {0})
64		elsni 4606	. . . . . . . . 9 ⊢ (𝑗 ∈ {0} → 𝑗 = 0)
65	63, 64	syl 18	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑗 = 0)
66	52	adantl 486	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 0 = (♯‘𝑑))
67	65, 66	eqtrd 2798	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → 𝑗 = (♯‘𝑑))
68		ccats1val2 14661	. . . . . . 7 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑗 = (♯‘𝑑)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = 𝑥)
69	47, 44, 67, 68	syl3anc 1398	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = 𝑥)
70		lswccats1 14668	. . . . . . 7 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
71	47, 44, 70	syl2anc 595	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
72	45, 69, 71	3brtr4d 5143	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 = ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
73	42	ad6antr 748	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → ∼ Er 𝐴)
74		simp-6r 799	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → 𝑑 ∈ ( ∼ Chain 𝐴))
75	74	chnwrd 18659	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → 𝑑 ∈ Word 𝐴)
76	75	adantr 485	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = (♯‘𝑑)) → 𝑑 ∈ Word 𝐴)
77		simp-6r 799	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = (♯‘𝑑)) → 𝑥 ∈ 𝐴)
78		simpr 489	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = (♯‘𝑑)) → 𝑗 = (♯‘𝑑))
79	76, 77, 78, 68	syl3anc 1398	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = (♯‘𝑑)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = 𝑥)
80		simp-4r 795	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥))
81		neneq 2964	. . . . . . . . . . . . 13 ⊢ (𝑑 ≠ ∅ → ¬ 𝑑 = ∅)
82	81	adantl 486	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → ¬ 𝑑 = ∅)
83	80, 82	orcnd 891	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → (lastS‘𝑑) ∼ 𝑥)
84	73, 83	ersym 8703	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → 𝑥 ∼ (lastS‘𝑑))
85	84	adantr 485	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = (♯‘𝑑)) → 𝑥 ∼ (lastS‘𝑑))
86	79, 85	eqbrtrd 5133	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 = (♯‘𝑑)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘𝑑))
87		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
88	87	breq1d 5119	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘𝑑)))
89		simpr 489	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) → ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑))
90	89	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑))
91		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑑 ∈ ( ∼ Chain 𝐴))
92	91	chnwrd 18659	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑑 ∈ Word 𝐴)
93		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑖 ∈ (0..^(♯‘𝑑)))
94		ccats1val1 14660	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = (𝑑‘𝑖))
95	92, 93, 94	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = (𝑑‘𝑖))
96	95	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → (𝑑‘𝑖) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖))
97	96	breq1d 5119	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
98	97	ralbidva 3186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) → (∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ∀𝑖 ∈ (0..^(♯‘𝑑))((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
99	98	ad6antr 748	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → (∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ∀𝑖 ∈ (0..^(♯‘𝑑))((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
100	90, 99	mpbid 235	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → ∀𝑖 ∈ (0..^(♯‘𝑑))((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑))
101		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))))
102		simp-5r 797	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → 𝑑 ∈ ( ∼ Chain 𝐴))
103	102	chnwrd 18659	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → 𝑑 ∈ Word 𝐴)
104	103, 49	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
105	104	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))) = (0..^((♯‘𝑑) + 1)))
106	101, 105	eleqtrd 2865	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → 𝑗 ∈ (0..^((♯‘𝑑) + 1)))
107	106	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → 𝑗 ∈ (0..^((♯‘𝑑) + 1)))
108		simp-7r 801	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → 𝑑 ∈ ( ∼ Chain 𝐴))
109	108	chnwrd 18659	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → 𝑑 ∈ Word 𝐴)
110		lencl 14566	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘𝑑) ∈ ℕ₀)
111		nn0uz 12895	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
112	111	eleq2i 2855	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑑) ∈ ℕ₀ ↔ (♯‘𝑑) ∈ (ℤ_≥‘0))
113	112	biimpi 219	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑑) ∈ ℕ₀ → (♯‘𝑑) ∈ (ℤ_≥‘0))
114	109, 110, 113	3syl 19	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → (♯‘𝑑) ∈ (ℤ_≥‘0))
115		fzosplitsni 13804	. . . . . . . . . . . 12 ⊢ ((♯‘𝑑) ∈ (ℤ_≥‘0) → (𝑗 ∈ (0..^((♯‘𝑑) + 1)) ↔ (𝑗 ∈ (0..^(♯‘𝑑)) ∨ 𝑗 = (♯‘𝑑))))
116	114, 115	syl 18	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → (𝑗 ∈ (0..^((♯‘𝑑) + 1)) ↔ (𝑗 ∈ (0..^(♯‘𝑑)) ∨ 𝑗 = (♯‘𝑑))))
117	107, 116	mpbid 235	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → (𝑗 ∈ (0..^(♯‘𝑑)) ∨ 𝑗 = (♯‘𝑑)))
118		df-ne 2959	. . . . . . . . . . 11 ⊢ (𝑗 ≠ (♯‘𝑑) ↔ ¬ 𝑗 = (♯‘𝑑))
119	118	bilani 509	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → ¬ 𝑗 = (♯‘𝑑))
120	117, 119	olcnd 890	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → 𝑗 ∈ (0..^(♯‘𝑑)))
121	88, 100, 120	rspcdva 3582	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) ∧ 𝑗 ≠ (♯‘𝑑)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘𝑑))
122	86, 121	pm2.61dane 3045	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘𝑑))
123	73, 122, 83	ertrd 8707	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ 𝑥)
124		simp-5r 797	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → 𝑥 ∈ 𝐴)
125	75, 124, 70	syl2anc 595	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
126	123, 125	breqtrrd 5139	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
127	72, 126	pm2.61dane 3045	. . . 4 ⊢ ((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
128	127	ralrimiva 3157	. . 3 ⊢ (((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) → ∀𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
129	8, 14, 24, 30, 31, 41, 128	chnind 18672	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) ∼ (lastS‘𝐶))
130		chner.3	. 2 ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))
131	2, 129, 130	rspcdva 3582	1 ⊢ (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∅c0 4286 {csn 4589 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 Er wer 8687 0cc0 11095 1c1 11096 + caddc 11098 ℕcn 12228 ℕ₀cn0 12499 ℤ_≥cuz 12857 ..^cfzo 13678 ♯chash 14362 Word cword 14546 lastSclsw 14595 ++ cconcat 14603 ⟨“cs1 14629 Chain cchn 18656

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 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

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 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-n0 12500 df-xnn0 12573 df-z 12587 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-hash 14363 df-word 14547 df-lsw 14596 df-concat 14604 df-s1 14630 df-substr 14675 df-pfx 14705 df-chn 18657

This theorem is referenced by: chnerlem2 47599

W3C validator