spthonepeq-av - Mathbox for Alexander van der Vekens

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Theorem spthonepeq-av 40958

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.)

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

**Proof of Theorem spthonepeq-av**
Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	spthonprop 40951	. 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 40914	. . . . . 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 1210	. . . . 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		simp1 1054	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → 𝐺 ∈ V)
6	5	anim1i 590	. . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
7		3anass 1035	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ↔ (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
8	6, 7	sylibr 223	. . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
9		issPth 40930	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(SPathS‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
10	8, 9	syl 17	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(SPathS‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
11	4, 10	anbi12d 743	. . . 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))))
12	1	wlkOnprop 40866	. . . . . . . 8 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
13		1wlkcl 40820	. . . . . . . . . . . . 13 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ₀)
14	1	1wlkp 40821	. . . . . . . . . . . . 13 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺))
15		df-f1 5809	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) ↔ (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃))
16		eqeq2 2621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 ↔ (𝑃‘0) = 𝐵))
17		eqtr3 2631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘(#‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → (𝑃‘(#‘𝐹)) = (𝑃‘0))
18		elnn0uz 11601	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐹) ∈ ℕ₀ ↔ (#‘𝐹) ∈ (ℤ_≥‘0))
19		eluzfz2 12220	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘0) → (#‘𝐹) ∈ (0...(#‘𝐹)))
20	18, 19	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘𝐹) ∈ (0...(#‘𝐹)))
21		0elfz 12305	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(#‘𝐹)))
22	20, 21	jca 553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((#‘𝐹) ∈ (0...(#‘𝐹)) ∧ 0 ∈ (0...(#‘𝐹))))
23		f1veqaeq 6418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) ∧ ((#‘𝐹) ∈ (0...(#‘𝐹)) ∧ 0 ∈ (0...(#‘𝐹)))) → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0))
24	22, 23	sylan2 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) ∧ (#‘𝐹) ∈ ℕ₀) → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0))
25	24	ex 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → ((#‘𝐹) ∈ ℕ₀ → ((𝑃‘(#‘𝐹)) = (𝑃‘0) → (#‘𝐹) = 0)))
26	25	com13 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘(#‘𝐹)) = (𝑃‘0) → ((#‘𝐹) ∈ ℕ₀ → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0)))
27	17, 26	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃‘(#‘𝐹)) = 𝐵 ∧ (𝑃‘0) = 𝐵) → ((#‘𝐹) ∈ ℕ₀ → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0)))
28	27	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) = 𝐵 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ₀ → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0))))
29	16, 28	syl6bi 242	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 = 𝐵 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ₀ → (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → (#‘𝐹) = 0)))))
30	29	com15 99	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
31	15, 30	sylbir 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ Fun ^◡𝑃) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
32	31	expcom 450	. . . . . . . . . . . . . 14 ⊢ (Fun ^◡𝑃 → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → ((#‘𝐹) ∈ ℕ₀ → (𝐴 = 𝐵 → (#‘𝐹) = 0))))))
33	32	com15 99	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) ∈ ℕ₀ → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun ^◡𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0))))))
34	13, 14, 33	sylc 63	. . . . . . . . . . . 12 ⊢ (𝐹(1Walks‘𝐺)𝑃 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (Fun ^◡𝑃 → (𝐴 = 𝐵 → (#‘𝐹) = 0)))))
35	34	3imp1 1272	. . . . . . . . . . 11 ⊢ (((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 → (#‘𝐹) = 0))
36		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 0 → (𝑃‘(#‘𝐹)) = (𝑃‘0))
37	36	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 0 → ((𝑃‘(#‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
38	37	anbi2d 736	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
39		eqtr2 2630	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
40	38, 39	syl6bi 242	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → 𝐴 = 𝐵))
41	40	com12 32	. . . . . . . . . . . . 13 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
42	41	3adant1 1072	. . . . . . . . . . . 12 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
43	42	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → ((#‘𝐹) = 0 → 𝐴 = 𝐵))
44	35, 43	impbid 201	. . . . . . . . . 10 ⊢ (((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
45	44	ex 449	. . . . . . . . 9 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
46	45	3ad2ant3 1077	. . . . . . . 8 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
47	12, 46	syl 17	. . . . . . 7 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → (Fun ^◡𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
48	47	adantld 482	. . . . . 6 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
49	48	adantr 480	. . . . 5 ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(TrailS‘𝐺)𝑃) → ((𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)))
50	49	imp 444	. . . 4 ⊢ (((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(TrailS‘𝐺)𝑃) ∧ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃)) → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))
51	11, 50	syl6bi 242	. . 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)))
52	51	3impia 1253	. 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))
53	2, 52	syl 17	1 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ^◡ccnv 5037 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ₀cn0 11169 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979 Vtxcvtx 25673 1Walksc1wlks 40796 WalksOncwlkson 40798 TrailSctrls 40899 TrailsOnctrlson 40900 SPathScspths 40920 SPathsOncspthson 40922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-spths 40924 df-spthson 40926

This theorem is referenced by: wspthsnonn0vne 41124

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