chnerlem1 - Mathbox for Ender Ting

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Theorem chnerlem1 48165

Description: In a chain constructed on an equivalence relation, the last element is equivalent to any. This theorem is a translation of chnub 18846 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 6916	. . 3 ⊢ (𝑖 = 𝐽 → (𝐶‘𝑖) = (𝐶‘𝐽))
2	1	breq1d 5118	. 2 ⊢ (𝑖 = 𝐽 → ((𝐶‘𝑖) ∼ (lastS‘𝐶) ↔ (𝐶‘𝐽) ∼ (lastS‘𝐶)))
3		fveq2 6916	. . . . 5 ⊢ (𝑐 = ∅ → (♯‘𝑐) = (♯‘∅))
4	3	oveq2d 7467	. . . 4 ⊢ (𝑐 = ∅ → (0..^(♯‘𝑐)) = (0..^(♯‘∅)))
5		fveq1 6915	. . . . 5 ⊢ (𝑐 = ∅ → (𝑐‘𝑖) = (∅‘𝑖))
6		fveq2 6916	. . . . 5 ⊢ (𝑐 = ∅ → (lastS‘𝑐) = (lastS‘∅))
7	5, 6	breq12d 5121	. . . 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 6916	. . . . 5 ⊢ (𝑐 = 𝑑 → (♯‘𝑐) = (♯‘𝑑))
10	9	oveq2d 7467	. . . 4 ⊢ (𝑐 = 𝑑 → (0..^(♯‘𝑐)) = (0..^(♯‘𝑑)))
11		fveq1 6915	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑖) = (𝑑‘𝑖))
12		fveq2 6916	. . . . 5 ⊢ (𝑐 = 𝑑 → (lastS‘𝑐) = (lastS‘𝑑))
13	11, 12	breq12d 5121	. . . 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 6916	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
16	15	breq1d 5118	. . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ (𝑐‘𝑗) ∼ (lastS‘𝑐)))
17	16	cbvralvw 3244	. . . 4 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑐))(𝑐‘𝑖) ∼ (lastS‘𝑐) ↔ ∀𝑗 ∈ (0..^(♯‘𝑐))(𝑐‘𝑗) ∼ (lastS‘𝑐))
18		fveq2 6916	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (♯‘𝑐) = (♯‘(𝑑 ++ ⟨“𝑥”⟩)))
19	18	oveq2d 7467	. . . . 5 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (0..^(♯‘𝑐)) = (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩))))
20		fveq1 6915	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (𝑐‘𝑗) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
21		fveq2 6916	. . . . . 6 ⊢ (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (lastS‘𝑐) = (lastS‘(𝑑 ++ ⟨“𝑥”⟩)))
22	20, 21	breq12d 5121	. . . . 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 6916	. . . . 5 ⊢ (𝑐 = 𝐶 → (♯‘𝑐) = (♯‘𝐶))
26	25	oveq2d 7467	. . . 4 ⊢ (𝑐 = 𝐶 → (0..^(♯‘𝑐)) = (0..^(♯‘𝐶)))
27		fveq1 6915	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
28		fveq2 6916	. . . . 5 ⊢ (𝑐 = 𝐶 → (lastS‘𝑐) = (lastS‘𝐶))
29	27, 28	breq12d 5121	. . . 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		hash0 14552	. . . . . . 7 ⊢ (♯‘∅) = 0
33		0nnn 12414	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
34	32, 33	eqneltri 2883	. . . . . 6 ⊢ ¬ (♯‘∅) ∈ ℕ
35		fzo0n0 13891	. . . . . 6 ⊢ ((0..^(♯‘∅)) ≠ ∅ ↔ (♯‘∅) ∈ ℕ)
36	34, 35	mtbir 326	. . . . 5 ⊢ ¬ (0..^(♯‘∅)) ≠ ∅
37		nne 2963	. . . . 5 ⊢ (¬ (0..^(♯‘∅)) ≠ ∅ ↔ (0..^(♯‘∅)) = ∅)
38	36, 37	mpbi 233	. . . 4 ⊢ (0..^(♯‘∅)) = ∅
39		rzal 4453	. . . 4 ⊢ ((0..^(♯‘∅)) = ∅ → ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅))
40	38, 39	mp1i 14	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘∅))(∅‘𝑖) ∼ (lastS‘∅))
41		chner.1	. . . . . . . 8 ⊢ (𝜑 → ∼ Er 𝐴)
42	41	ad6antr 749	. . . . . . 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)
43		simp-5r 798	. . . . . . 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)
44	42, 43	erref 8776	. . . . . 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)
45		simp-6r 800	. . . . . . . 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))
46	45	chnwrd 18832	. . . . . . 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)
47		simplr 781	. . . . . . . . . . 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”⟩))))
48		ccatws1len 14810	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘(𝑑 ++ ⟨“𝑥”⟩)) = ((♯‘𝑑) + 1))
49	46, 48	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))
50		fveq2 6916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = ∅ → (♯‘𝑑) = (♯‘∅))
51	50, 32	eqtr2di 2815	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ∅ → 0 = (♯‘𝑑))
52	51	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ∅ → (♯‘𝑑) = 0)
53	52	adantl 487	. . . . . . . . . . . . . . 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)
54	53	oveq1d 7466	. . . . . . . . . . . . . 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))
55		0p1e1 12503	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
56	54, 55	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)
57	49, 56	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)
58	57	oveq2d 7467	. . . . . . . . . . 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))
59	47, 58	eleqtrd 2866	. . . . . . . . . 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))
60		fzo01 13922	. . . . . . . . . 10 ⊢ (0..^1) = {0}
61	59, 60	eleqtrdi 2874	. . . . . . . . 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})
62	61	elsnd 4602	. . . . . . . 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)
63	51	adantl 487	. . . . . . . 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))
64	62, 63	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))
65		ccats1val2 14817	. . . . . . 7 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑗 = (♯‘𝑑)) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) = 𝑥)
66	46, 43, 64, 65	syl3anc 1399	. . . . . 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)
67		lswccats1 14824	. . . . . . 7 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑥 ∈ 𝐴) → (lastS‘(𝑑 ++ ⟨“𝑥”⟩)) = 𝑥)
68	46, 43, 67	syl2anc 596	. . . . . 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)
69	44, 66, 68	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”⟩)))
70	41	ad6antr 749	. . . . . . 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)
71		simp-6r 800	. . . . . . . . . . . 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))
72	71	chnwrd 18832	. . . . . . . . . . 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)
73	72	adantr 486	. . . . . . . . . 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)
74		simp-6r 800	. . . . . . . . . 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)
75		simpr 490	. . . . . . . . . 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))
76	73, 74, 75, 65	syl3anc 1399	. . . . . . . . 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)
77		simp-4r 796	. . . . . . . . . . . 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))
78		neneq 2965	. . . . . . . . . . . . 13 ⊢ (𝑑 ≠ ∅ → ¬ 𝑑 = ∅)
79	78	adantl 487	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) ∼ 𝑥)) ∧ ∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑)) ∧ 𝑗 ∈ (0..^(♯‘(𝑑 ++ ⟨“𝑥”⟩)))) ∧ 𝑑 ≠ ∅) → ¬ 𝑑 = ∅)
80	77, 79	orcnd 892	. . . . . . . . . . 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)
81	70, 80	ersym 8768	. . . . . . . . . 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))
82	81	adantr 486	. . . . . . . . 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))
83	76, 82	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))
84		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗))
85	84	breq1d 5118	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑗) ∼ (lastS‘𝑑)))
86		simp-4r 796	. . . . . . . . . 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))
87		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑑 ∈ ( ∼ Chain 𝐴))
88	87	chnwrd 18832	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑑 ∈ Word 𝐴)
89		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → 𝑖 ∈ (0..^(♯‘𝑑)))
90		ccats1val1 14816	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ Word 𝐴 ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = (𝑑‘𝑖))
91	88, 89, 90	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) = (𝑑‘𝑖))
92	91	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → (𝑑‘𝑖) = ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖))
93	92	breq1d 5118	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) ∧ 𝑖 ∈ (0..^(♯‘𝑑))) → ((𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
94	93	ralbidva 3187	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ( ∼ Chain 𝐴)) → (∀𝑖 ∈ (0..^(♯‘𝑑))(𝑑‘𝑖) ∼ (lastS‘𝑑) ↔ ∀𝑖 ∈ (0..^(♯‘𝑑))((𝑑 ++ ⟨“𝑥”⟩)‘𝑖) ∼ (lastS‘𝑑)))
95	94	ad6antr 749	. . . . . . . . . 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)))
96	86, 95	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))
97		simpr 490	. . . . . . . . . . . . 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”⟩))))
98		simp-5r 798	. . . . . . . . . . . . . . . 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))
99	98	chnwrd 18832	. . . . . . . . . . . . . . 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)
100	99, 48	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))
101	100	oveq2d 7467	. . . . . . . . . . . . 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)))
102	97, 101	eleqtrd 2866	. . . . . . . . . . . 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)))
103	102	ad2antrr 739	. . . . . . . . . . 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)))
104		simp-7r 802	. . . . . . . . . . . . . 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))
105	104	chnwrd 18832	. . . . . . . . . . . . 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)
106		lencl 14719	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Word 𝐴 → (♯‘𝑑) ∈ ℕ₀)
107		elnn0uz 13046	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑑) ∈ ℕ₀ ↔ (♯‘𝑑) ∈ (ℤ_≥‘0))
108	107	biimpi 219	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑑) ∈ ℕ₀ → (♯‘𝑑) ∈ (ℤ_≥‘0))
109	105, 106, 108	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))
110		fzosplitsni 13954	. . . . . . . . . . . 12 ⊢ ((♯‘𝑑) ∈ (ℤ_≥‘0) → (𝑗 ∈ (0..^((♯‘𝑑) + 1)) ↔ (𝑗 ∈ (0..^(♯‘𝑑)) ∨ 𝑗 = (♯‘𝑑))))
111	109, 110	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))))
112	103, 111	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)))
113		df-ne 2960	. . . . . . . . . . 11 ⊢ (𝑗 ≠ (♯‘𝑑) ↔ ¬ 𝑗 = (♯‘𝑑))
114	113	bilani 510	. . . . . . . . . 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))
115	112, 114	olcnd 891	. . . . . . . . 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)))
116	85, 96, 115	rspcdva 3581	. . . . . . . 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))
117	83, 116	pm2.61dane 3046	. . . . . . 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))
118	70, 117, 80	ertrd 8772	. . . . . 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)
119		simp-5r 798	. . . . . . 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)
120	72, 119, 67	syl2anc 596	. . . . . 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)
121	118, 120	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”⟩)))
122	69, 121	pm2.61dane 3046	. . . 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”⟩)))
123	122	ralrimiva 3158	. . 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”⟩)))
124	8, 14, 24, 30, 31, 40, 123	chnind 18845	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) ∼ (lastS‘𝐶))
125		chner.3	. 2 ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))
126	2, 124, 125	rspcdva 3581	1 ⊢ (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∅c0 4281 {csn 4588 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 Er wer 8752 0cc0 11239 1c1 11240 + caddc 11242 ℕcn 12375 ℕ₀cn0 12646 ℤ_≥cuz 13005 ..^cfzo 13828 ♯chash 14515 Word cword 14699 lastSclsw 14748 ++ cconcat 14756 ⟨“cs1 14783 Chain cchn 18829

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-lsw 14749 df-concat 14757 df-s1 14784 df-substr 14831 df-pfx 14863 df-chn 18830

This theorem is used by: chnerlem2 48166

W3C validator