chnerlem1 - Mathbox for Ender Ting

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Theorem chnerlem1 47162

Description: In a chain constructed on an equivalence relation, the last element is equivalent to any. This theorem is a translation of chnub 18549 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 6835	. . 3 ⊢ (𝑖 = 𝐽 → (𝐶‘𝑖) = (𝐶‘𝐽))
2	1	breq1d 5109	. 2 ⊢ (𝑖 = 𝐽 → ((𝐶‘𝑖) ∼ (lastS‘𝐶) ↔ (𝐶‘𝐽) ∼ (lastS‘𝐶)))
3		fveq2 6835	. . . . 5 ⊢ (𝑐 = ∅ → (♯‘𝑐) = (♯‘∅))
4	3	oveq2d 7376	. . . 4 ⊢ (𝑐 = ∅ → (0..^(♯‘𝑐)) = (0..^(♯‘∅)))
5		fveq1 6834	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐‘𝑖) = (∅‘𝑖))
6		fveq2 6835	. . . . 5 ⊢ (𝑐 = ∅ → (lastS‘𝑐) = (lastS‘∅))
7	5, 6	breq12d 5112	. . . 4 ⊢ (𝑐 = ∅ → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (∅‘𝑖) ∼ (lastS‘∅)))
8	4, 7	raleqbidv 3317	. . 3 ⊢ (𝑐 = ∅ → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅)))
9		fveq2 6835	. . . . 5 ⊢ (𝑐 = 𝑑 → (♯‘𝑐) = (♯‘𝑑))
10	9	oveq2d 7376	. . . 4 ⊢ (𝑐 = 𝑑 → (0..^(♯‘𝑐)) = (0..^(♯‘𝑑)))
11		fveq1 6834	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑖) = (𝑑‘𝑖))
12		fveq2 6835	. . . . 5 ⊢ (𝑐 = 𝑑 → (lastS‘𝑐) = (lastS‘𝑑))
13	11, 12	breq12d 5112	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝑑‘𝑖) ∼ (lastS‘𝑑)))
14	10, 13	raleqbidv 3317	. . 3 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)))
15		fveq2 6835	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
16	15	breq1d 5109	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝑐‘𝑗) ∼ (lastS‘𝑐)))
17	16	cbvralvw 3215	. . . 4 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^(♯‘𝑐))(𝑐‘𝑗) ∼ (lastS‘𝑐))
18		fveq2 6835	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (♯‘𝑐) = (♯‘(𝑑 ++ ⟨“𝑥”⟩)))
19	18	oveq2d 7376	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (0..^(♯‘𝑐)) = (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))))
20		fveq1 6834	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (𝑐‘𝑗) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
21		fveq2 6835	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (lastS‘𝑐) = (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
22	20, 21	breq12d 5112	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → ((𝑐‘𝑗) ∼ (lastS‘𝑐) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
23	19, 22	raleqbidv 3317	. . . 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 283	. . 3 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘(𝑑 ++ ⟨“𝑥”⟩))))
25		fveq2 6835	. . . . 5 ⊢ (𝑐 = 𝐶 → (♯‘𝑐) = (♯‘𝐶))
26	25	oveq2d 7376	. . . 4 ⊢ (𝑐 = 𝐶 → (0..^(♯‘𝑐)) = (0..^(♯‘𝐶)))
27		fveq1 6834	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
28		fveq2 6835	. . . . 5 ⊢ (𝑐 = 𝐶 → (lastS‘𝑐) = (lastS‘𝐶))
29	27, 28	breq12d 5112	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝐶‘𝑖) ∼ (lastS‘𝐶)))
30	26, 29	raleqbidv 3317	. . 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 12185	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
33		hash0 14294	. . . . . . . 8 ⊢ (♯‘∅) = 0
34	33	eleq1i 2828	. . . . . . 7 ⊢ ((♯‘∅) ∈ ℕ ↔ 0 ∈ ℕ)
35	32, 34	mtbir 323	. . . . . 6 ⊢ ¬ (♯‘∅) ∈ ℕ
36		fzo0n0 13636	. . . . . 6 ⊢ ((0..^(♯‘∅)) ≠ ∅ ↔ (♯‘∅) ∈ ℕ)
37	35, 36	mtbir 323	. . . . 5 ⊢ ¬ (0..^(♯‘∅)) ≠ ∅
38		nne 2937	. . . . 5 ⊢ (¬ (0..^(♯‘∅)) ≠ ∅ ↔ (0..^(♯‘∅)) = ∅)
39	37, 38	mpbi 230	. . . 4 ⊢ (0..^(♯‘∅)) = ∅
40		rzal 4448	. . . 4 ⊢ ((0..^(♯‘∅)) = ∅ → ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅))
41	39, 40	mp1i 13	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅))
42		chner.1	. . . . . . . 8 ⊢ (𝜑 → ∼ Er 𝐴)
43	42	ad6antr 737	. . . . . . 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 786	. . . . . . 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 8658	. . . . . 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 788	. . . . . . . 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 18535	. . . . . . 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 769	. . . . . . . . . . 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 14548	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
50	47, 49	syl 17	. . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
52	51, 33	eqtr2di 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ∅ → 0 = (♯‘𝑑))
53	52	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ∅ → (♯‘𝑑) = 0)
54	53	adantl 481	. . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . 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 12266	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
57	55, 56	eqtrdi 2788	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 7376	. . . . . . . . . . 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 2839	. . . . . . . . . 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 13667	. . . . . . . . . . 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 2839	. . . . . . . . 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 4598	. . . . . . . . 9 ⊢ (𝑗 ∈ {0} → 𝑗 = 0)
65	63, 64	syl 17	. . . . . . . 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 481	. . . . . . . 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 2772	. . . . . . 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 14555	. . . . . . 7 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑗 = (♯‘𝑑)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = 𝑥)
69	47, 44, 67, 68	syl3anc 1374	. . . . . 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 14562	. . . . . . 7 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
71	47, 44, 70	syl2anc 585	. . . . . 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 5131	. . . . 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 737	. . . . . . 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 788	. . . . . . . . . . . 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 18535	. . . . . . . . . . 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 480	. . . . . . . . . 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 788	. . . . . . . . . 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 484	. . . . . . . . . 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 1374	. . . . . . . . 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 784	. . . . . . . . . . . 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 2939	. . . . . . . . . . . . 13 ⊢ (𝑑 ≠ ∅ → ¬ 𝑑 = ∅)
82	81	adantl 481	. . . . . . . . . . . 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 879	. . . . . . . . . . 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 8650	. . . . . . . . . 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 480	. . . . . . . . 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 5121	. . . . . . . 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 6835	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
88	87	breq1d 5109	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘𝑑)))
89		simpr 484	. . . . . . . . . . 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 731	. . . . . . . . . 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 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑑 ∈ ( ∼ Chain 𝐴))
92	91	chnwrd 18535	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑑 ∈ Word 𝐴)
93		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑖 ∈ (0..^(♯‘𝑑)))
94		ccats1val1 14554	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = (𝑑‘𝑖))
95	92, 93, 94	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = (𝑑‘𝑖))
96	95	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → (𝑑‘𝑖) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖))
97	96	breq1d 5109	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
98	97	ralbidva 3158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) → (∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ∀𝑖 ∈ (0..^(♯‘𝑑))((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
99	98	ad6antr 737	. . . . . . . . . 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 232	. . . . . . . . 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 484	. . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . 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 18535	. . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 790	. . . . . . . . . . . . . 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 18535	. . . . . . . . . . . . 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 14460	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘𝑑) ∈ ℕ₀)
111		nn0uz 12793	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
112	111	eleq2i 2829	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑑) ∈ ℕ₀ ↔ (♯‘𝑑) ∈ (ℤ_≥‘0))
113	112	biimpi 216	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑑) ∈ ℕ₀ → (♯‘𝑑) ∈ (ℤ_≥‘0))
114	109, 110, 113	3syl 18	. . . . . . . . . . . 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 13699	. . . . . . . . . . . 12 ⊢ ((♯‘𝑑) ∈ (ℤ_≥‘0) → (𝑗 ∈ (0..^((♯‘𝑑) + 1)) ↔ (𝑗 ∈ (0..^(♯‘𝑑)) ∨ 𝑗 = (♯‘𝑑))))
116	114, 115	syl 17	. . . . . . . . . . 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 232	. . . . . . . . . 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		simpr 484	. . . . . . . . . . 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 ≠ (♯‘𝑑))
119		df-ne 2934	. . . . . . . . . . 11 ⊢ (𝑗 ≠ (♯‘𝑑) ↔ ¬ 𝑗 = (♯‘𝑑))
120	118, 119	sylib 218	. . . . . . . . . 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))
121	117, 120	olcnd 878	. . . . . . . . 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)))
122	88, 100, 121	rspcdva 3578	. . . . . . . 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))
123	86, 122	pm2.61dane 3020	. . . . . . 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))
124	73, 123, 83	ertrd 8654	. . . . . 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)
125		simp-5r 786	. . . . . . 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)
126	75, 125, 70	syl2anc 585	. . . . . 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)
127	124, 126	breqtrrd 5127	. . . . 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”⟩)))
128	72, 127	pm2.61dane 3020	. . . 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”⟩)))
129	128	ralrimiva 3129	. . 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”⟩)))
130	8, 14, 24, 30, 31, 41, 129	chnind 18548	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) ∼ (lastS‘𝐶))
131		chner.3	. 2 ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))
132	2, 130, 131	rspcdva 3578	1 ⊢ (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∅c0 4286 {csn 4581 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 Er wer 8634 0cc0 11030 1c1 11031 + caddc 11033 ℕcn 12149 ℕ₀cn0 12405 ℤ_≥cuz 12755 ..^cfzo 13574 ♯chash 14257 Word cword 14440 lastSclsw 14489 ++ cconcat 14497 ⟨“cs1 14523 Chain cchn 18532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-n0 12406 df-xnn0 12479 df-z 12493 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-lsw 14490 df-concat 14498 df-s1 14524 df-substr 14569 df-pfx 14599 df-chn 18533

This theorem is referenced by: chnerlem2 47163

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