Theorem usgra2adedgwlk 26142

Description: In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Hypotheses

Ref	Expression
usgra2adedgspth.f	⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩}
usgra2adedgspth.p	⊢ 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}

Assertion

Ref	Expression
usgra2adedgwlk	⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))))

Proof of Theorem usgra2adedgwlk

Step	Hyp	Ref	Expression
1		usgrav 25867	. . . . 5 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2	1	adantr 480	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3		usgraedgrnv 25906	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
4	3	ancomd 466	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
5	4	adantrr 749	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
6	5	simprd 478	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐴 ∈ 𝑉)
7	3	adantrr 749	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
8	7	simprd 478	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐵 ∈ 𝑉)
9		usgraedgrnv 25906	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
10	9	adantrl 748	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
11	10	simprd 478	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → 𝐶 ∈ 𝑉)
12	6, 8, 11	3jca 1235	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
13	2, 12	jca 553	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
14		simpl 472	. . . . 5 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
15		usgra2adedgspthlem1 26139	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
16	15	adantl 481	. . . . 5 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))) → ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
17		fvex 6113	. . . . . . 7 ⊢ (◡𝐸‘{𝐴, 𝐵}) ∈ V
18		fvex 6113	. . . . . . 7 ⊢ (◡𝐸‘{𝐵, 𝐶}) ∈ V
19	17, 18	pm3.2i 470	. . . . . 6 ⊢ ((◡𝐸‘{𝐴, 𝐵}) ∈ V ∧ (◡𝐸‘{𝐵, 𝐶}) ∈ V)
20		usgra2adedgspth.f	. . . . . 6 ⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩}
21		usgra2adedgspth.p	. . . . . 6 ⊢ 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}
22	19, 20, 21	constr2wlk 26128	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}) → 𝐹(𝑉 Walks 𝐸)𝑃))
23	14, 16, 22	sylc 63	. . . 4 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))) → 𝐹(𝑉 Walks 𝐸)𝑃)
24	19, 20	2trllemA 26080	. . . . 5 ⊢ (#‘𝐹) = 2
25	24	a1i 11	. . . 4 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))) → (#‘𝐹) = 2)
26	21	2wlklemA 26084	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴)
27	26	eqcomd 2616	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = (𝑃‘0))
28	21	2wlklemB 26085	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝑃‘1) = 𝐵)
29	28	eqcomd 2616	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (𝑃‘1))
30	21	2wlklemC 26086	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑉 → (𝑃‘2) = 𝐶)
31	30	eqcomd 2616	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝑃‘2))
32	27, 29, 31	3anim123i 1240	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))
33	32	adantl 481	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))
34	33	adantr 480	. . . 4 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))) → (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))
35	23, 25, 34	3jca 1235	. . 3 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
36	13, 35	mpancom 700	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
37	36	ex 449	1 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   USGrph cusg 25859   Walks cwalk 26026

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-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-rmo 2904  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-cda 8873  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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036

This theorem is referenced by:  usgra2adedgwlkonALT  26144  usg2wlk  26145