eupatrl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > eupatrl		Structured version Visualization version GIF version

Theorem eupatrl 26495

Description: An Eulerian path is a trail.

Unfortunately, the edge function 𝐹 of an Eulerian path has the domain (1...(#‘𝐹)), whereas the edge functions of all kinds of walks defined here have the domain (0..^(#‘𝐹)) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 26162, fargshiftfv 26163, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

**Hypothesis**
Ref	Expression
eupatrl.f	⊢ 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))

**Assertion**
Ref	Expression
eupatrl	⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝐺(𝑉 Trails 𝐸)𝑃)

Distinct variable groups: 𝑥,𝐸 𝑥,𝐹 𝑥,𝑃 𝑥,𝑉 Allowed substitution hint: 𝐺(𝑥)

Proof of Theorem eupatrl Dummy variables 𝑒 𝑓 𝑘 𝑛 𝑝 𝑣 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eupa 26490	. . . 4 ⊢ EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ₀ (𝑓:(1...𝑛)–1-1-onto→dom 𝑒 ∧ 𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓‘𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝‘𝑘)}))})
2	1	brovmpt2ex 7236	. . 3 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3		eupatrl.f	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
4		ovex 6577	. . . . . . . 8 ⊢ (0..^(#‘𝐹)) ∈ V
5	4	mptex 6390	. . . . . . 7 ⊢ (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) ∈ V
6	3, 5	eqeltri 2684	. . . . . 6 ⊢ 𝐺 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝐹 ∈ V → 𝐺 ∈ V)
8	7	anim1i 590	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐺 ∈ V ∧ 𝑃 ∈ V))
9	8	anim2i 591	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)))
10	2, 9	syl 17	. 2 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)))
11		f1ofn 6051	. . . . . . . . . . . . 13 ⊢ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 → 𝐹 Fn (1...𝑛))
12		fseq1hash 13026	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑛)) → (#‘𝐹) = 𝑛)
13	11, 12	sylan2 490	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝐹:(1...𝑛)–1-1-onto→dom 𝐸) → (#‘𝐹) = 𝑛)
14	13	ancoms 468	. . . . . . . . . . 11 ⊢ ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑛 ∈ ℕ₀) → (#‘𝐹) = 𝑛)
15		f1of1 6049	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → 𝐹:(1...(#‘𝐹))–1-1→dom 𝐸)
16	3	fargshiftf1 26165	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)
17	16	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(1...(#‘𝐹))–1-1→dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸))
18	15, 17	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸))
19	18	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)
20		f1ofo 6057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → 𝐹:(1...(#‘𝐹))–onto→dom 𝐸)
21	3	fargshiftfo 26166	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)
22	21	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(1...(#‘𝐹))–onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸))
23	20, 22	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸))
24	23	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)
25		df-f1o 5811	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸))
26	19, 24, 25	sylanbrc 695	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → 𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸)
27		f1ofn 6051	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 → 𝐺 Fn (0..^(#‘𝐹)))
28		hashfn 13025	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 Fn (0..^(#‘𝐹)) → (#‘𝐺) = (#‘(0..^(#‘𝐹))))
29	28	anim2i 591	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐺 Fn (0..^(#‘𝐹))) → ((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))))
30	29	ancoms 468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 Fn (0..^(#‘𝐹)) ∧ (#‘𝐹) ∈ ℕ₀) → ((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))))
31	27, 30	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → ((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))))
32		hashfzo0 13077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
33	32	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((#‘𝐺) = (#‘(0..^(#‘𝐹))) ↔ (#‘𝐺) = (#‘𝐹)))
34	33	biimpa 500	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))) → (#‘𝐺) = (#‘𝐹))
35		pm3.2 462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐺) = (#‘𝐹) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹))))
36	35	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → ((#‘𝐺) = (#‘𝐹) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹))))
37	34, 36	syl5com 31	. . . . . . . . . . . . . . . 16 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))) → ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹))))
38	31, 37	mpcom 37	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)))
39	26, 38	sylancom 698	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)))
40		df-f1 5809	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun ^◡𝐺))
41		iswrdi 13164	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺:(0..^(#‘𝐹))⟶dom 𝐸 → 𝐺 ∈ Word dom 𝐸)
42	41	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun ^◡𝐺) → 𝐺 ∈ Word dom 𝐸)
43	40, 42	sylbi 206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 → 𝐺 ∈ Word dom 𝐸)
44	43	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸) → 𝐺 ∈ Word dom 𝐸)
45	25, 44	sylbi 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 → 𝐺 ∈ Word dom 𝐸)
46	45	ad4antr 764	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → 𝐺 ∈ Word dom 𝐸)
47	40	simprbi 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 → Fun ^◡𝐺)
48	47	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸) → Fun ^◡𝐺)
49	25, 48	sylbi 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 → Fun ^◡𝐺)
50	49	ad4antr 764	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → Fun ^◡𝐺)
51	46, 50	jca 553	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺))
52		eqcom 2617	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐺) = (#‘𝐹) ↔ (#‘𝐹) = (#‘𝐺))
53	52	biimpi 205	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝐺) = (#‘𝐹) → (#‘𝐹) = (#‘𝐺))
54	53	ad3antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → (#‘𝐹) = (#‘𝐺))
55	54	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → (0...(#‘𝐹)) = (0...(#‘𝐺)))
56	55	feq2d 5944	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...(#‘𝐺))⟶𝑉))
57	56	biimpa 500	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃:(0...(#‘𝐺))⟶𝑉)
58		f1of 6050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → 𝐹:(1...(#‘𝐹))⟶dom 𝐸)
59		fargshiftlem 26162	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (𝑙 + 1) ∈ (1...(#‘𝐹)))
60		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑙 + 1) ∈ (1...(#‘𝐹)))
61		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 = (𝑙 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑙 + 1)))
62	61	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑘 = (𝑙 + 1) → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
64		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑘 = (𝑙 + 1) → (𝑘 − 1) = ((𝑙 + 1) − 1))
65		elfzoelz 12339	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑙 ∈ (0..^(#‘𝐹)) → 𝑙 ∈ ℤ)
66	65	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑙 ∈ (0..^(#‘𝐹)) → 𝑙 ∈ ℂ)
67	66	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → 𝑙 ∈ ℂ)
68		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → 1 ∈ ℂ)
69	67, 68	pncand 10272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → ((𝑙 + 1) − 1) = 𝑙)
70	64, 69	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝑘 − 1) = 𝑙)
71	70	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝑃‘(𝑘 − 1)) = (𝑃‘𝑙))
72		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑘 = (𝑙 + 1) → (𝑃‘𝑘) = (𝑃‘(𝑙 + 1)))
73	72	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝑃‘𝑘) = (𝑃‘(𝑙 + 1)))
74	71, 73	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})
75	63, 74	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
76	75	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
77	76	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) → (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
78	77	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
79	78	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
80		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) → (#‘𝐹) ∈ ℕ₀)
81	80	anim1i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → ((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸))
82	81	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸))
83		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → 𝑙 ∈ (0..^(#‘𝐹)))
84	83	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → 𝑙 ∈ (0..^(#‘𝐹)))
85	3	fargshiftfv 26163	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑙 ∈ (0..^(#‘𝐹)) → (𝐺‘𝑙) = (𝐹‘(𝑙 + 1))))
86	85	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (𝐺‘𝑙) = (𝐹‘(𝑙 + 1)))
87	86	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (𝐹‘(𝑙 + 1)) = (𝐺‘𝑙))
88	82, 84, 87	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐹‘(𝑙 + 1)) = (𝐺‘𝑙))
89	88	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐸‘(𝐹‘(𝑙 + 1))) = (𝐸‘(𝐺‘𝑙)))
90	89	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))} ↔ (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
91	79, 90	bitrd 267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
92	60, 91	rspcdv 3285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
93	92	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
94	93	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹)))) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
95	59, 94	mpancom 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
96	95	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝐹) ∈ ℕ₀ → (𝑙 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))))
97	96	com24 93	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝐹) ∈ ℕ₀ → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑙 ∈ (0..^(#‘𝐹)) → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))))
98	97	imp31 447	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → (𝑙 ∈ (0..^(#‘𝐹)) → (𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
99	98	ralrimiv 2948	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})
100	99	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
101	100	expcom 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
102	58, 101	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
103	102	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
104		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝐺) = (#‘𝐹) → (0..^(#‘𝐺)) = (0..^(#‘𝐹)))
105	104	raleqdv 3121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐺) = (#‘𝐹) → (∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))} ↔ ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
106	105	imbi2d 329	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝐺) = (#‘𝐹) → ((∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}) ↔ (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
107	103, 106	syl5ibr 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐺) = (#‘𝐹) → ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
108	107	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) → ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
109	108	imp31 447	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})
110	109	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})
111	51, 57, 110	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
112	111	exp31 628	. . . . . . . . . . . . . 14 ⊢ (((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))))
113	39, 112	mpancom 700	. . . . . . . . . . . . 13 ⊢ ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))))
114	113	a1i 11	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 𝑛 → ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))))
115		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (#‘𝐹) → (1...𝑛) = (1...(#‘𝐹)))
116	115	eqcoms 2618	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 𝑛 → (1...𝑛) = (1...(#‘𝐹)))
117		f1oeq2 6041	. . . . . . . . . . . . . 14 ⊢ ((1...𝑛) = (1...(#‘𝐹)) → (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ↔ 𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸))
118	116, 117	syl 17	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 𝑛 → (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ↔ 𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸))
119		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (#‘𝐹) → (𝑛 ∈ ℕ₀ ↔ (#‘𝐹) ∈ ℕ₀))
120	119	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 𝑛 → (𝑛 ∈ ℕ₀ ↔ (#‘𝐹) ∈ ℕ₀))
121	118, 120	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 𝑛 → ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑛 ∈ ℕ₀) ↔ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ₀)))
122	116	raleqdv 3121	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 𝑛 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}))
123		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (#‘𝐹) → (0...𝑛) = (0...(#‘𝐹)))
124	123	eqcoms 2618	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 𝑛 → (0...𝑛) = (0...(#‘𝐹)))
125	124	feq2d 5944	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 𝑛 → (𝑃:(0...𝑛)⟶𝑉 ↔ 𝑃:(0...(#‘𝐹))⟶𝑉))
126	125	imbi1d 330	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 𝑛 → ((𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})) ↔ (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))))
127	122, 126	imbi12d 333	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 𝑛 → ((∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))) ↔ (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))))
128	114, 121, 127	3imtr4d 282	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 𝑛 → ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑛 ∈ ℕ₀) → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))))
129	14, 128	mpcom 37	. . . . . . . . . 10 ⊢ ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑛 ∈ ℕ₀) → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))))
130	129	ex 449	. . . . . . . . 9 ⊢ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 → (𝑛 ∈ ℕ₀ → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))))
131	130	com24 93	. . . . . . . 8 ⊢ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 → (𝑃:(0...𝑛)⟶𝑉 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} → (𝑛 ∈ ℕ₀ → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))))
132	131	3imp 1249	. . . . . . 7 ⊢ ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → (𝑛 ∈ ℕ₀ → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
133	132	com12 32	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
134	133	rexlimiv 3009	. . . . 5 ⊢ (∃𝑛 ∈ ℕ₀ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
135	134	adantl 481	. . . 4 ⊢ ((𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ₀ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)})) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))}))
136	135	a1i 11	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ₀ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)})) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
137		eqid 2610	. . . 4 ⊢ dom 𝐸 = dom 𝐸
138		iseupa 26492	. . . 4 ⊢ (dom 𝐸 = dom 𝐸 → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ₀ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}))))
139	137, 138	mp1i 13	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ₀ (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}))))
140		istrl 26067	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → (𝐺(𝑉 Trails 𝐸)𝑃 ↔ ((𝐺 ∈ Word dom 𝐸 ∧ Fun ^◡𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺‘𝑙)) = {(𝑃‘𝑙), (𝑃‘(𝑙 + 1))})))
141	136, 139, 140	3imtr4d 282	. 2 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝐺(𝑉 Trails 𝐸)𝑃))
142	10, 141	mpcom 37	1 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝐺(𝑉 Trails 𝐸)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 UMGrph cumg 25841 Trails ctrail 26027 EulPaths ceup 26489

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-umgra 25842 df-wlk 26036 df-trail 26037 df-eupa 26490

This theorem is referenced by: (None)

W3C validator