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Theorem spthonepeq 28021

Description: The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)

**Assertion**
Ref	Expression
spthonepeq	⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))

**Proof of Theorem spthonepeq**
Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	spthonprop 28014	. 2 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃)))
3	1	istrlson 27976	. . . . . 6 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
4	3	3adantl1 1164	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
5		isspth 27993	. . . . . 6 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))
6	5	a1i 11	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
7	4, 6	anbi12d 630	. . . 4 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃) ↔ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃) ∧ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))))
8	1	wlkonprop 27928	. . . . . . . 8 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
9		wlkcl 27885	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
10	1	wlkp 27886	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
11		df-f1 6423	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ↔ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃))
12		eqeq2 2750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 ↔ (𝑃‘0) = 𝐵))
13		eqtr3 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘(♯‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → (𝑃‘(♯‘𝐹)) = (𝑃‘0))
14		elnn0uz 12552	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (ℤ_≥‘0))
15		eluzfz2 13193	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
16	14, 15	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
17		0elfz 13282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
18	16, 17	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹))))
19		f1veqaeq 7111	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ∧ ((♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)))) → ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → (♯‘𝐹) = 0))
20	18, 19	sylan2 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ₀) → ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → (♯‘𝐹) = 0))
21	20	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → ((♯‘𝐹) ∈ ℕ₀ → ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → (♯‘𝐹) = 0)))
22	21	com13 88	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → (♯‘𝐹) = 0)))
23	13, 22	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃‘(♯‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → (♯‘𝐹) = 0)))
24	23	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) = 𝐵 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → (♯‘𝐹) = 0))))
25	12, 24	syl6bi 252	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → (♯‘𝐹) = 0)))))
26	25	com15 101	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (♯‘𝐹) = 0)))))
27	11, 26	sylbir 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃) → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (♯‘𝐹) = 0)))))
28	27	expcom 413	. . . . . . . . . . . . . 14 ⊢ (Fun ^◡𝑃 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (♯‘𝐹) = 0))))))
29	28	com15 101	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → (Fun ^◡𝑃 → (𝐴 = 𝐵 → (♯‘𝐹) = 0))))))
30	9, 10, 29	sylc 65	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → (Fun ^◡𝑃 → (𝐴 = 𝐵 → (♯‘𝐹) = 0)))))
31	30	3imp1 1345	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 → (♯‘𝐹) = 0))
32		fveqeq2 6765	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
33	32	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
34		eqtr2 2762	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
35	33, 34	syl6bi 252	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → 𝐴 = 𝐵))
36	35	com12 32	. . . . . . . . . . . . 13 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → 𝐴 = 𝐵))
37	36	3adant1 1128	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → 𝐴 = 𝐵))
38	37	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → ((♯‘𝐹) = 0 → 𝐴 = 𝐵))
39	31, 38	impbid 211	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))
40	39	ex 412	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
41	40	3ad2ant3 1133	. . . . . . . 8 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
42	8, 41	syl 17	. . . . . . 7 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
43	42	adantld 490	. . . . . 6 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
44	43	adantr 480	. . . . 5 ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃) → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
45	44	imp 406	. . . 4 ⊢ (((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃) ∧ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃)) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))
46	7, 45	syl6bi 252	. . 3 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
47	46	3impia 1115	. 2 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃)) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))
48	2, 47	syl 17	1 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 Vcvv 3422 class class class wbr 5070 ^◡ccnv 5579 Fun wfun 6412 ⟶wf 6414 –1-1→wf1 6415 ‘cfv 6418 (class class class)co 7255 0cc0 10802 ℕ₀cn0 12163 ℤ_≥cuz 12511 ...cfz 13168 ♯chash 13972 Vtxcvtx 27269 Walkscwlks 27866 WalksOncwlkson 27867 Trailsctrls 27960 TrailsOnctrlson 27961 SPathscspths 27982 SPathsOncspthson 27984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-wlks 27869 df-wlkson 27870 df-trls 27962 df-trlson 27963 df-pths 27985 df-spths 27986 df-spthson 27988

This theorem is referenced by: wspthsnonn0vne 28183

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