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Theorem pthdivtx 26625

Description: The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)

**Assertion**
Ref	Expression
pthdivtx	⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))

**Proof of Theorem pthdivtx**
Step	Hyp	Ref	Expression
1		ispth 26619	. . 3 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅))
2		trliswlk 26594	. . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2622	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 26512	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺))
5		elfz0lmr 12583	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...(#‘𝐹)) → (𝐽 = 0 ∨ 𝐽 ∈ (1..^(#‘𝐹)) ∨ 𝐽 = (#‘𝐹)))
6		elfzo1 12517	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) ↔ (𝐼 ∈ ℕ ∧ (#‘𝐹) ∈ ℕ ∧ 𝐼 < (#‘𝐹)))
7		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝐹) ∈ ℕ → (#‘𝐹) ∈ ℕ₀)
8	7	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐼 ∈ ℕ ∧ (#‘𝐹) ∈ ℕ ∧ 𝐼 < (#‘𝐹)) → (#‘𝐹) ∈ ℕ₀)
9	6, 8	sylbi 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℕ₀)
10	9	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (#‘𝐹) ∈ ℕ₀)
11		fvinim0ffz 12587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (#‘𝐹) ∈ ℕ₀) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))))))
12	10, 11	sylan2 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))))))
13		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = 0 → (𝑃‘𝐽) = (𝑃‘0))
14	13	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = 0 → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
15	14	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘0)))
16		ffun 6048	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
17	16	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → Fun 𝑃)
18		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(#‘𝐹)))
19		fzo0ss1 12498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
20		fzossfz 12488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
21	19, 20	sstri 3612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (1..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
22	21	sseli 3599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) → 𝐼 ∈ (0...(#‘𝐹)))
23		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑃 = (0...(#‘𝐹)) → (𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ (0...(#‘𝐹))))
24	22, 23	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (dom 𝑃 = (0...(#‘𝐹)) → (𝐼 ∈ (1..^(#‘𝐹)) → 𝐼 ∈ dom 𝑃))
25	18, 24	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (𝐼 ∈ (1..^(#‘𝐹)) → 𝐼 ∈ dom 𝑃))
26	25	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → 𝐼 ∈ dom 𝑃)
27	17, 26	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
28	27	adantrl 752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
29		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → 𝐼 ∈ (1..^(#‘𝐹)))
30		funfvima 6492	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃) → (𝐼 ∈ (1..^(#‘𝐹)) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹)))))
31	28, 29, 30	sylc 65	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))))
32		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝐼) = (𝑃‘0) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹)))))
33	31, 32	syl5ibcom 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘0) → (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹)))))
34	15, 33	sylbid 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹)))))
35		nnel 2906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘0) ∈ (𝑃 “ (1..^(#‘𝐹))))
36	34, 35	syl6ibr 242	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹)))))
37	36	necon2ad 2809	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
38	37	adantrd 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
39	12, 38	sylbid 230	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
40	39	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
41	40	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
42	41	a1d 25	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
43	42	3imp 1256	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
44	43	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
45	44	a1d 25	. . . . . . . . . . 11 ⊢ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
46	45	ex 450	. . . . . . . . . 10 ⊢ (𝐽 = 0 → (𝐼 ∈ (1..^(#‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
47		fvres 6207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (1..^(#‘𝐹)) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = (𝑃‘𝐼))
48	47	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = (𝑃‘𝐼))
49	48	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = (𝑃‘𝐼))
50	49	eqcomd 2628	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝑃‘𝐼) = ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼))
51		fvres 6207	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (1..^(#‘𝐹)) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽) = (𝑃‘𝐽))
52	51	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽) = (𝑃‘𝐽))
53	52	eqcomd 2628	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝑃‘𝐽) = ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽))
54	50, 53	eqeq12d 2637	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽)))
55		fssres 6070	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (1..^(#‘𝐹)) ⊆ (0...(#‘𝐹))) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺))
56	21, 55	mpan2 707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺))
57		df-f1 5893	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺) ↔ ((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹)))))
58	57	biimpri 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹)))) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺))
59	56, 58	sylan 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹)))) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺))
60	59	3adant3 1081	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺))
61		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))))
62	61	ancomd 467	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (1..^(#‘𝐹))))
63		f1veqaeq 6514	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ↾ (1..^(#‘𝐹))):(1..^(#‘𝐹))–1-1→(Vtx‘𝐺) ∧ (𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (1..^(#‘𝐹)))) → (((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽) → 𝐼 = 𝐽))
64	60, 62, 63	syl2an2r 876	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 ↾ (1..^(#‘𝐹)))‘𝐼) = ((𝑃 ↾ (1..^(#‘𝐹)))‘𝐽) → 𝐼 = 𝐽))
65	54, 64	sylbid 230	. . . . . . . . . . . . . . 15 ⊢ (((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) ∧ (𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → 𝐼 = 𝐽))
66	65	ancoms 469	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))) ∧ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)) → ((𝑃‘𝐼) = (𝑃‘𝐽) → 𝐼 = 𝐽))
67	66	necon3d 2815	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))) ∧ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)) → (𝐼 ≠ 𝐽 → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
68	67	ex 450	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝐼 ≠ 𝐽 → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
69	68	com23 86	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (1..^(#‘𝐹)) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
70	69	ex 450	. . . . . . . . . 10 ⊢ (𝐽 ∈ (1..^(#‘𝐹)) → (𝐼 ∈ (1..^(#‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
71	9	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (#‘𝐹) ∈ ℕ₀)
72	71, 11	sylan2 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ ↔ ((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))))))
73		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐽 = (#‘𝐹) → (𝑃‘𝐽) = (𝑃‘(#‘𝐹)))
74	73	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = (#‘𝐹) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(#‘𝐹))))
75	74	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) ↔ (𝑃‘𝐼) = (𝑃‘(#‘𝐹))))
76	27	adantrl 752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
77		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → 𝐼 ∈ (1..^(#‘𝐹)))
78	76, 77, 30	sylc 65	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))))
79		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝐼) = (𝑃‘(#‘𝐹)) → ((𝑃‘𝐼) ∈ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹)))))
80	78, 79	syl5ibcom 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘(#‘𝐹)) → (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹)))))
81	75, 80	sylbid 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹)))))
82		nnel 2906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))) ↔ (𝑃‘(#‘𝐹)) ∈ (𝑃 “ (1..^(#‘𝐹))))
83	81, 82	syl6ibr 242	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘𝐼) = (𝑃‘𝐽) → ¬ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹)))))
84	83	necon2ad 2809	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → ((𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
85	84	adantld 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃‘0) ∉ (𝑃 “ (1..^(#‘𝐹))) ∧ (𝑃‘(#‘𝐹)) ∉ (𝑃 “ (1..^(#‘𝐹)))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
86	72, 85	sylbid 230	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ (𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹)))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
87	86	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
88	87	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
89	88	a1d 25	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
90	89	3imp 1256	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
91	90	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
92	91	a1d 25	. . . . . . . . . . 11 ⊢ ((𝐽 = (#‘𝐹) ∧ 𝐼 ∈ (1..^(#‘𝐹))) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))))
93	92	ex 450	. . . . . . . . . 10 ⊢ (𝐽 = (#‘𝐹) → (𝐼 ∈ (1..^(#‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
94	46, 70, 93	3jaoi 1391	. . . . . . . . 9 ⊢ ((𝐽 = 0 ∨ 𝐽 ∈ (1..^(#‘𝐹)) ∨ 𝐽 = (#‘𝐹)) → (𝐼 ∈ (1..^(#‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
95	5, 94	syl 17	. . . . . . . 8 ⊢ (𝐽 ∈ (0...(#‘𝐹)) → (𝐼 ∈ (1..^(#‘𝐹)) → (𝐼 ≠ 𝐽 → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
96	95	3imp21 1277	. . . . . . 7 ⊢ ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
97	96	com12 32	. . . . . 6 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
98	97	3exp 1264	. . . . 5 ⊢ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
99	2, 4, 98	3syl 18	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))))
100	99	3imp 1256	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
101	1, 100	sylbi 207	. 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)))
102	101	imp 445	1 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {cpr 4179 class class class wbr 4653 ^◡ccnv 5113 dom cdm 5114 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 < clt 10074 ℕcn 11020 ℕ₀cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Vtxcvtx 25874 Walkscwlks 26492 Trailsctrls 26587 Pathscpths 26608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495 df-trls 26589 df-pths 26612

This theorem is referenced by: pthdadjvtx 26626 upgr4cycl4dv4e 27045

W3C validator