pthhashvtx - Metamath Proof Explorer

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

Theorem pthhashvtx 30320

Description: A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)

**Hypothesis**
Ref	Expression
pthhashvtx.1	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
pthhashvtx	⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉))

**Proof of Theorem pthhashvtx**
Step	Hyp	Ref	Expression
1		hashfz0 14580	. . . 4 ⊢ (((♯‘𝐹) − 1) ∈ ℕ₀ → (♯‘(0...((♯‘𝐹) − 1))) = (((♯‘𝐹) − 1) + 1))
2		pthiswlk 30315	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		wlkcl 30201	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
4	2, 3	syl 18	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
5		nn0cn 12619	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℂ)
6		npcan1 11744	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℂ → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
7	4, 5, 6	3syl 19	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
8	1, 7	sylan9eqr 2817	. . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → (♯‘(0...((♯‘𝐹) − 1))) = (♯‘𝐹))
9		pthhashvtx.1	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
10	9	wlkp 30202	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
11	2, 10	syl 18	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
12	11	ffnd 6710	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝑃 Fn (0...(♯‘𝐹)))
13		fzfi 14118	. . . . 5 ⊢ (0...((♯‘𝐹) − 1)) ∈ Fin
14		resfnfinfin 9329	. . . . 5 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ (0...((♯‘𝐹) − 1)) ∈ Fin) → (𝑃 ↾ (0...((♯‘𝐹) − 1))) ∈ Fin)
15	12, 13, 14	sylancl 598	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (0...((♯‘𝐹) − 1))) ∈ Fin)
16		simpr 490	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → ((♯‘𝐹) − 1) ∈ ℕ₀)
17		fzssp1 13704	. . . . . . . 8 ⊢ (0...((♯‘𝐹) − 1)) ⊆ (0...(((♯‘𝐹) − 1) + 1))
18	7	oveq2d 7436	. . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → (0...(((♯‘𝐹) − 1) + 1)) = (0...(♯‘𝐹)))
19	17, 18	sseqtrid 3950	. . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → (0...((♯‘𝐹) − 1)) ⊆ (0...(♯‘𝐹)))
20	11, 19	fssresd 6749	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (0...((♯‘𝐹) − 1))):(0...((♯‘𝐹) − 1))⟶𝑉)
21	20	adantr 486	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → (𝑃 ↾ (0...((♯‘𝐹) − 1))):(0...((♯‘𝐹) − 1))⟶𝑉)
22		fz1ssfz0 13760	. . . . . . . . . 10 ⊢ (1...((♯‘𝐹) − 1)) ⊆ (0...((♯‘𝐹) − 1))
23	22	a1i 11	. . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → (1...((♯‘𝐹) − 1)) ⊆ (0...((♯‘𝐹) − 1)))
24	20, 23	fssresd 6749	. . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))):(1...((♯‘𝐹) − 1))⟶𝑉)
25		ispth 30311	. . . . . . . . . 10 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
26	25	simp2bi 1164	. . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))))
27		nn0z 12720	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
28		fzoval 13797	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℤ → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1)))
29	27, 28	syl 18	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1)))
30	4, 29	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐹(Paths‘𝐺)𝑃 → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1)))
31	30	reseq2d 5969	. . . . . . . . . . . 12 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (1..^(♯‘𝐹))) = (𝑃 ↾ (1...((♯‘𝐹) − 1))))
32		resabs1 5996	. . . . . . . . . . . . 13 ⊢ ((1...((♯‘𝐹) − 1)) ⊆ (0...((♯‘𝐹) − 1)) → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))) = (𝑃 ↾ (1...((♯‘𝐹) − 1))))
33	22, 32	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))) = (𝑃 ↾ (1...((♯‘𝐹) − 1)))
34	31, 33	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (1..^(♯‘𝐹))) = ((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))))
35	34	cnveqd 5852	. . . . . . . . . 10 ⊢ (𝐹(Paths‘𝐺)𝑃 → ^◡(𝑃 ↾ (1..^(♯‘𝐹))) = ^◡((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))))
36	35	funeqd 6561	. . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → (Fun ^◡(𝑃 ↾ (1..^(♯‘𝐹))) ↔ Fun ^◡((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1)))))
37	26, 36	mpbid 235	. . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → Fun ^◡((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))))
38		df-f1 6543	. . . . . . . 8 ⊢ (((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))):(1...((♯‘𝐹) − 1))–1-1→𝑉 ↔ (((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))):(1...((♯‘𝐹) − 1))⟶𝑉 ∧ Fun ^◡((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1)))))
39	24, 37, 38	sylanbrc 595	. . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))):(1...((♯‘𝐹) − 1))–1-1→𝑉)
40	39	adantr 486	. . . . . 6 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))):(1...((♯‘𝐹) − 1))–1-1→𝑉)
41	38	simprbi 503	. . . . . 6 ⊢ (((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))):(1...((♯‘𝐹) − 1))–1-1→𝑉 → Fun ^◡((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))))
42	40, 41	syl 18	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → Fun ^◡((𝑃 ↾ (0...((♯‘𝐹) − 1))) ↾ (1...((♯‘𝐹) − 1))))
43		snsspr1 4769	. . . . . . . 8 ⊢ {0} ⊆ {0, (♯‘𝐹)}
44		imass2 6054	. . . . . . . 8 ⊢ ({0} ⊆ {0, (♯‘𝐹)} → (𝑃 “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)}))
45	43, 44	ax-mp 5	. . . . . . 7 ⊢ (𝑃 “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)})
46		0elfz 13761	. . . . . . . . 9 ⊢ (((♯‘𝐹) − 1) ∈ ℕ₀ → 0 ∈ (0...((♯‘𝐹) − 1)))
47	46	snssd 4743	. . . . . . . 8 ⊢ (((♯‘𝐹) − 1) ∈ ℕ₀ → {0} ⊆ (0...((♯‘𝐹) − 1)))
48		resima2 6056	. . . . . . . 8 ⊢ ({0} ⊆ (0...((♯‘𝐹) − 1)) → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) = (𝑃 “ {0}))
49		sseq1 3933	. . . . . . . 8 ⊢ (((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) = (𝑃 “ {0}) → (((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)}) ↔ (𝑃 “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)})))
50	47, 48, 49	3syl 19	. . . . . . 7 ⊢ (((♯‘𝐹) − 1) ∈ ℕ₀ → (((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)}) ↔ (𝑃 “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)})))
51	45, 50	mpbiri 261	. . . . . 6 ⊢ (((♯‘𝐹) − 1) ∈ ℕ₀ → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)}))
52		resima2 6056	. . . . . . . . . 10 ⊢ ((1...((♯‘𝐹) − 1)) ⊆ (0...((♯‘𝐹) − 1)) → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1))) = (𝑃 “ (1...((♯‘𝐹) − 1))))
53	22, 52	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1))) = (𝑃 “ (1...((♯‘𝐹) − 1)))
54	30	imaeq2d 6051	. . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 “ (1..^(♯‘𝐹))) = (𝑃 “ (1...((♯‘𝐹) − 1))))
55	53, 54	eqtr4id 2814	. . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1))) = (𝑃 “ (1..^(♯‘𝐹))))
56	55	ineq2d 4117	. . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1)))) = ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))))
57	25	simp3bi 1165	. . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)
58	56, 57	eqtrd 2795	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 “ {0, (♯‘𝐹)}) ∩ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1)))) = ∅)
59		ssdisj 4412	. . . . . 6 ⊢ ((((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) ⊆ (𝑃 “ {0, (♯‘𝐹)}) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1)))) = ∅) → (((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) ∩ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1)))) = ∅)
60	51, 58, 59	syl2anr 609	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → (((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ {0}) ∩ ((𝑃 ↾ (0...((♯‘𝐹) − 1))) “ (1...((♯‘𝐹) − 1)))) = ∅)
61	16, 21, 42, 60	f1resfz0f1d 13930	. . . 4 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → (𝑃 ↾ (0...((♯‘𝐹) − 1))):(0...((♯‘𝐹) − 1))–1-1→𝑉)
62	9	fvexi 6899	. . . . 5 ⊢ 𝑉 ∈ V
63		hashf1dmcdm 14592	. . . . 5 ⊢ (((𝑃 ↾ (0...((♯‘𝐹) − 1))) ∈ Fin ∧ 𝑉 ∈ V ∧ (𝑃 ↾ (0...((♯‘𝐹) − 1))):(0...((♯‘𝐹) − 1))–1-1→𝑉) → (♯‘(0...((♯‘𝐹) − 1))) ≤ (♯‘𝑉))
64	62, 63	mp3an2 1478	. . . 4 ⊢ (((𝑃 ↾ (0...((♯‘𝐹) − 1))) ∈ Fin ∧ (𝑃 ↾ (0...((♯‘𝐹) − 1))):(0...((♯‘𝐹) − 1))–1-1→𝑉) → (♯‘(0...((♯‘𝐹) − 1))) ≤ (♯‘𝑉))
65	15, 61, 64	syl2an2r 698	. . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → (♯‘(0...((♯‘𝐹) − 1))) ≤ (♯‘𝑉))
66	8, 65	eqbrtrrd 5128	. 2 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ((♯‘𝐹) − 1) ∈ ℕ₀) → (♯‘𝐹) ≤ (♯‘𝑉))
67		0nn0m1nnn0 12756	. . . . 5 ⊢ ((♯‘𝐹) = 0 ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ ¬ ((♯‘𝐹) − 1) ∈ ℕ₀))
68	67	biimpri 231	. . . 4 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ ¬ ((♯‘𝐹) − 1) ∈ ℕ₀) → (♯‘𝐹) = 0)
69	4, 68	sylan 592	. . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ¬ ((♯‘𝐹) − 1) ∈ ℕ₀) → (♯‘𝐹) = 0)
70		hashge0 14534	. . . 4 ⊢ (𝑉 ∈ V → 0 ≤ (♯‘𝑉))
71	62, 70	ax-mp 5	. . 3 ⊢ 0 ≤ (♯‘𝑉)
72	69, 71	eqbrtrdi 5143	. 2 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ ¬ ((♯‘𝐹) − 1) ∈ ℕ₀) → (♯‘𝐹) ≤ (♯‘𝑉))
73	66, 72	pm2.61dan 825	1 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⊆ wss 3895 ∩ cin 4089 ∅c0 4278 {csn 4584 {cpr 4586 class class class wbr 5102 ^◡ccnv 5649 ↾ cres 5652 “ cima 5653 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 –1-1→wf1 6535 ‘cfv 6538 (class class class)co 7420 Fincfn 8976 ℂcc 11201 0cc0 11203 1c1 11204 + caddc 11206 ≤ cle 11347 − cmin 11544 ℕ₀cn0 12609 ℤcz 12696 ...cfz 13642 ..^cfzo 13791 ♯chash 14477 Vtxcvtx 29579 Walkscwlks 30182 Trailsctrls 30278 Pathscpths 30300

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-oadd 8483 df-er 8720 df-map 8852 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-dju 9985 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-n0 12610 df-xnn0 12683 df-z 12697 df-uz 12969 df-fz 13643 df-fzo 13792 df-hash 14478 df-word 14662 df-wlks 30185 df-trls 30280 df-pths 30304

This theorem is used by: usgrcyclgt2v 35986 acycgr1v 35990

W3C validator