pthdadjvtx - Metamath Proof Explorer

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

Theorem pthdadjvtx 29811

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 13702	. . 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 29808	. . . . . . . . . . 11 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
4		wlkcl 29699	. . . . . . . . . . 11 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
5		1zzd 12549	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 ∈ ℤ)
6		nn0z 12539	. . . . . . . . . . . . . . 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 13611	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (1..^(♯‘𝐹)) ↔ (1 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ ∧ 1 < (♯‘𝐹)))
10	5, 7, 8, 9	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 ∈ (1..^(♯‘𝐹)))
11		0elfz 13569	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
12	11	adantr 480	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 0 ∈ (0...(♯‘𝐹)))
13		ax-1ne0 11098	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 0
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 < (♯‘𝐹)) → 1 ≠ 0)
15	10, 12, 14	3jca 1129	. . . . . . . . . . . 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 29810	. . . . . . . . 9 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (1 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 1 ≠ 0)) → (𝑃‘1) ≠ (𝑃‘0))
20	2, 18, 19	syl2anc 585	. . . . . . . 8 ⊢ ((1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘1) ≠ (𝑃‘0))
21	20	necomd 2988	. . . . . . 7 ⊢ ((1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘1))
22	21	3adant1 1131	. . . . . 6 ⊢ ((𝐼 = 0 ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘1))
23		fveq2 6834	. . . . . . . 8 ⊢ (𝐼 = 0 → (𝑃‘𝐼) = (𝑃‘0))
24		fv0p1e1 12290	. . . . . . . 8 ⊢ (𝐼 = 0 → (𝑃‘(𝐼 + 1)) = (𝑃‘1))
25	23, 24	neeq12d 2994	. . . . . . 7 ⊢ (𝐼 = 0 → ((𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1)))
26	25	3ad2ant1 1134	. . . . . 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 1120	. . . 4 ⊢ (𝐼 = 0 → (1 < (♯‘𝐹) → (𝐹(Paths‘𝐺)𝑃 → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))))
29		simp3 1139	. . . . . 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 13635	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
32	31	sseli 3918	. . . . . . . . 9 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ (0..^(♯‘𝐹)))
33		fzofzp1 13710	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^(♯‘𝐹)) → (𝐼 + 1) ∈ (0...(♯‘𝐹)))
34	32, 33	syl 17	. . . . . . . 8 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 + 1) ∈ (0...(♯‘𝐹)))
35		elfzoelz 13604	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ ℤ)
36	35	zcnd 12625	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 𝐼 ∈ ℂ)
37		1cnd 11130	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 1 ∈ ℂ)
38	13	a1i 11	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → 1 ≠ 0)
39	36, 37, 38	3jca 1129	. . . . . . . . 9 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0))
40		addn0nid 11561	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → (𝐼 + 1) ≠ 𝐼)
41	40	necomd 2988	. . . . . . . . 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 1129	. . . . . . 7 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (𝐼 ∈ (1..^(♯‘𝐹)) ∧ (𝐼 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ (𝐼 + 1)))
44	43	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝐼 ∈ (1..^(♯‘𝐹)) ∧ (𝐼 + 1) ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ (𝐼 + 1)))
45		pthdivtx 29810	. . . . . 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 585	. . . . 5 ⊢ ((𝐼 ∈ (1..^(♯‘𝐹)) ∧ 1 < (♯‘𝐹) ∧ 𝐹(Paths‘𝐺)𝑃) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))
47	46	3exp 1120	. . . 4 ⊢ (𝐼 ∈ (1..^(♯‘𝐹)) → (1 < (♯‘𝐹) → (𝐹(Paths‘𝐺)𝑃 → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))))
48	28, 47	jaoi 858	. . 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 < clt 11170 ℕ₀cn0 12428 ℤcz 12515 ...cfz 13452 ..^cfzo 13599 ♯chash 14283 Walkscwlks 29680 Pathscpths 29793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-wlks 29683 df-trls 29774 df-pths 29797

This theorem is referenced by: 2pthnloop 29814 upgr3v3e3cycl 30265 upgr4cycl4dv4e 30270

W3C validator