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Theorem spthonepeq 29820

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	spthonprop 29813	. 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 29773	. . . . . 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 1168	. . . . 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 29790	. . . . . 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 633	. . . 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 29725	. . . . . . . 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 29684	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
10	1	wlkp 29685	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
11		df-f1 6503	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ↔ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃))
12		eqeq2 2748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 ↔ (𝑃‘0) = 𝐵))
13		eqtr3 2758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘(♯‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → (𝑃‘(♯‘𝐹)) = (𝑃‘0))
14		elnn0uz 12829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (ℤ_≥‘0))
15		eluzfz2 13486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
16	14, 15	sylbi 217	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
17		0elfz 13578	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
18	16, 17	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹))))
19		f1veqaeq 7211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺) ∧ ((♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)))) → ((𝑃‘(♯‘𝐹)) = (𝑃‘0) → (♯‘𝐹) = 0))
20	18, 19	sylan2 594	. . . . . . . . . . . . . . . . . . . . . 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	biimtrdi 253	. . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 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 1349	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 → (♯‘𝐹) = 0))
32		fveqeq2 6849	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
33	32	anbi2d 631	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
34		eqtr2 2757	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
35	33, 34	biimtrdi 253	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 212	. . . . . . . . . 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 1136	. . . . . . . 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	biimtrdi 253	. . 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 1118	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3429 class class class wbr 5085 ^◡ccnv 5630 Fun wfun 6492 ⟶wf 6494 –1-1→wf1 6495 ‘cfv 6498 (class class class)co 7367 0cc0 11038 ℕ₀cn0 12437 ℤ_≥cuz 12788 ...cfz 13461 ♯chash 14292 Vtxcvtx 29065 Walkscwlks 29665 WalksOncwlkson 29666 Trailsctrls 29757 TrailsOnctrlson 29758 SPathscspths 29779 SPathsOncspthson 29781

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-wlks 29668 df-wlkson 29669 df-trls 29759 df-trlson 29760 df-spths 29783 df-spthson 29785

This theorem is referenced by: wspthsnonn0vne 29985

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