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Theorem spthonepeq 29838

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 2739	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	spthonprop 29831	. 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 29791	. . . . . 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 1173	. . . . 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 29808	. . . . . 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 638	. . . 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 29743	. . . . . . . 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 29702	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
10	1	wlkp 29703	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
11		df-f1 6490	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ↔ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃))
12		eqeq2 2751	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 ↔ (𝑃‘0) = 𝐵))
13		eqtr3 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘(♯‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → (𝑃‘(♯‘𝐹)) = (𝑃‘0))
14		elnn0uz 12820	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (ℤ_≥‘0))
15		eluzfz2 13477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
16	14, 15	sylbi 218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
17		0elfz 13569	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
18	16, 17	jca 516	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹))))
19		f1veqaeq 7200	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ∧ ((♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)))) → ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → (♯‘𝐹) = 0))
20	18, 19	sylan2 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ₀) → ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → (♯‘𝐹) = 0))
21	20	ex 413	. . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) = 𝐵 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) → (♯‘𝐹) = 0))))
25	12, 24	biimtrdi 254	. . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃) → ((𝑃‘0) = 𝐴 → ((𝑃‘(♯‘𝐹)) = 𝐵 → ((♯‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (♯‘𝐹) = 0)))))
28	27	expcom 414	. . . . . . . . . . . . . 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 1354	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 → (♯‘𝐹) = 0))
32		fveqeq2 6836	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
33	32	anbi2d 636	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
34		eqtr2 2760	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
35	33, 34	biimtrdi 254	. . . . . . . . . . . . . 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 1136	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → 𝐴 = 𝐵))
38	37	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → ((♯‘𝐹) = 0 → 𝐴 = 𝐵))
39	31, 38	impbid 213	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))
40	39	ex 413	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
41	40	3ad2ant3 1141	. . . . . . . 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 491	. . . . . 6 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
44	43	adantr 481	. . . . 5 ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃) → ((𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0)))
45	44	imp 407	. . . 4 ⊢ (((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃) ∧ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃)) → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))
46	7, 45	biimtrdi 254	. . 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 1123	. 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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 Vcvv 3431 class class class wbr 5072 ^◡ccnv 5617 Fun wfun 6479 ⟶wf 6481 –1-1→wf1 6482 ‘cfv 6485 (class class class)co 7356 0cc0 11029 ℕ₀cn0 12428 ℤ_≥cuz 12779 ...cfz 13452 ♯chash 14283 Vtxcvtx 29083 Walkscwlks 29683 WalksOncwlkson 29684 Trailsctrls 29775 TrailsOnctrlson 29776 SPathscspths 29797 SPathsOncspthson 29799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 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 208 df-an 397 df-or 854 df-ifp 1069 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 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 29686 df-wlkson 29687 df-trls 29777 df-trlson 29778 df-spths 29801 df-spthson 29803

This theorem is referenced by: wspthsnonn0vne 30003

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