pthdadjvtx - Metamath Proof Explorer

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Theorem pthdadjvtx 29766

Description: The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)

**Assertion**
Ref	Expression
pthdadjvtx	⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 𝐼 ∈ (0..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))

**Proof of Theorem pthdadjvtx**
Step	Hyp	Ref	Expression
1		elfzo0l 13806	. . 3 ⊢ (𝐼 ∈ (0..^(♯‘𝐹)) → (𝐼 = 0 ∨ 𝐼 ∈ (1..^(♯‘𝐹))))
2		simpr 484	. . . . . . . . 9 ⊢ ((1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → 𝐹(Paths‘𝐺)𝑃)
3		pthiswlk 29763	. . . . . . . . . . 11 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
4		wlkcl 29651	. . . . . . . . . . 11 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
5		1zzd 12674	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 ∈ ℤ)
6		nn0z 12664	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
7	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → (♯‘𝐹) ∈ ℤ)
8		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 < (♯‘𝐹))
9		fzolb 13722	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (1..^(♯‘𝐹)) ↔ (1 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ ∧ 1 < (♯‘𝐹)))
10	5, 7, 8, 9	syl3anbrc 1343	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 ∈ (1..^(♯‘𝐹)))
11		0elfz 13681	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
12	11	adantr 480	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 0 ∈ (0...(♯‘𝐹)))
13		ax-1ne0 11253	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 0
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 ≠ 0)
15	10, 12, 14	3jca 1128	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → (1 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 1 ≠ 0))
16	15	ex 412	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1 < (♯‘𝐹) → (1 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 1 ≠ 0)))
17	3, 4, 16	3syl 18	. . . . . . . . . 10 ⊢ (𝐹(Paths‘𝐺)𝑃 → (1 < (♯‘𝐹) → (1 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 1 ≠ 0)))
18	17	impcom 407	. . . . . . . . 9 ⊢ ((1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (1 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 1 ≠ 0))
19		pthdivtx 29765	. . . . . . . . 9 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (1 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 1 ≠ 0)) → (𝑃‘1) ≠ (𝑃‘0))
20	2, 18, 19	syl2anc 583	. . . . . . . 8 ⊢ ((1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘1) ≠ (𝑃‘0))
21	20	necomd 3002	. . . . . . 7 ⊢ ((1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘1))
22	21	3adant1 1130	. . . . . 6 ⊢ ((𝐼 = 0 ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘1))
23		fveq2 6920	. . . . . . . 8 ⊢ (𝐼 = 0 → (𝑃‘𝐼) = (𝑃‘0))
24		fv0p1e1 12416	. . . . . . . 8 ⊢ (𝐼 = 0 → (𝑃‘(𝐼 + 1)) = (𝑃‘1))
25	23, 24	neeq12d 3008	. . . . . . 7 ⊢ (𝐼 = 0 → ((𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1)))
26	25	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐼 = 0 ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → ((𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1)))
27	22, 26	mpbird 257	. . . . 5 ⊢ ((𝐼 = 0 ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))
28	27	3exp 1119	. . . 4 ⊢ (𝐼 = 0 → (1 < (♯‘𝐹) → (𝐹(Paths‘𝐺)𝑃 → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))))
29		simp3 1138	. . . . . 6 ⊢ ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → 𝐹(Paths‘𝐺)𝑃)
30		id 22	. . . . . . . 8 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (1..^(♯‘𝐹)))
31		fzo0ss1 13746	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
32	31	sseli 4004	. . . . . . . . 9 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0..^(♯‘𝐹)))
33		fzofzp1 13814	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^(♯‘𝐹)) → (𝐼 + 1) ∈ (0...(♯‘𝐹)))
34	32, 33	syl 17	. . . . . . . 8 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 + 1) ∈ (0...(♯‘𝐹)))
35		elfzoelz 13716	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ ℤ)
36	35	zcnd 12748	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ ℂ)
37		1cnd 11285	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 1 ∈ ℂ)
38	13	a1i 11	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 1 ≠ 0)
39	36, 37, 38	3jca 1128	. . . . . . . . 9 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0))
40		addn0nid 11710	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → (𝐼 + 1) ≠ 𝐼)
41	40	necomd 3002	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → 𝐼 ≠ (𝐼 + 1))
42	39, 41	syl 17	. . . . . . . 8 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ≠ (𝐼 + 1))
43	30, 34, 42	3jca 1128	. . . . . . 7 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) ∧ (𝐼 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ (𝐼 + 1)))
44	43	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝐼 ∈ (1..^(♯‘𝐹)) ∧ (𝐼 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ (𝐼 + 1)))
45		pthdivtx 29765	. . . . . 6 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ (𝐼 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ (𝐼 + 1))) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))
46	29, 44, 45	syl2anc 583	. . . . 5 ⊢ ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))
47	46	3exp 1119	. . . 4 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (1 < (♯‘𝐹) → (𝐹(Paths‘𝐺)𝑃 → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))))
48	28, 47	jaoi 856	. . 3 ⊢ ((𝐼 = 0 ∨ 𝐼 ∈ (1..^(♯‘𝐹))) → (1 < (♯‘𝐹) → (𝐹(Paths‘𝐺)𝑃 → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))))
49	1, 48	syl 17	. 2 ⊢ (𝐼 ∈ (0..^(♯‘𝐹)) → (1 < (♯‘𝐹) → (𝐹(Paths‘𝐺)𝑃 → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))))
50	49	3imp31 1112	1 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 𝐼 ∈ (0..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 ℕ₀cn0 12553 ℤcz 12639 ...cfz 13567 ..^cfzo 13711 ♯chash 14379 Walkscwlks 29632 Pathscpths 29748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-wlks 29635 df-trls 29728 df-pths 29752

This theorem is referenced by: 2pthnloop 29767 upgr3v3e3cycl 30212 upgr4cycl4dv4e 30217

W3C validator