Theorem elwwlks2ons3 29935

Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)

Hypothesis

Ref	Expression
wwlks2onv.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
elwwlks2ons3	⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏   𝑊,𝑏

Proof of Theorem elwwlks2ons3

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
2		wwlks2onv.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	elwwlks2ons3im 29934	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
4		anass 468	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
5	1, 3, 4	sylanbrc 583	. . 3 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
6		simpr 484	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
7		s3eq2 14812	. . . . . 6 ⊢ (𝑏 = (𝑊‘1) → ⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩)
8		eqeq2 2741	. . . . . . 7 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ↔ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩))
9		eleq1 2816	. . . . . . 7 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
10	8, 9	anbi12d 632	. . . . . 6 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
11	7, 10	syl 17	. . . . 5 ⊢ (𝑏 = (𝑊‘1) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
12	11	adantl 481	. . . 4 ⊢ ((((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ∧ 𝑏 = (𝑊‘1)) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
13		simpr 484	. . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
14		eleq1 2816	. . . . . . 7 ⊢ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
15	14	biimpac 478	. . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
16	13, 15	jca 511	. . . . 5 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
17	16	adantr 480	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
18	6, 12, 17	rspcedvd 3587	. . 3 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
19	5, 18	syl 17	. 2 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
20		eleq1 2816	. . . . 5 ⊢ (⟨“𝐴𝑏𝐶”⟩ = 𝑊 → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
21	20	eqcoms 2737	. . . 4 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
22	21	biimpa 476	. . 3 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
23	22	rexlimivw 3130	. 2 ⊢ (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
24	19, 23	impbii 209	1 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ‘cfv 6499  (class class class)co 7369  1c1 11045  2c2 12217  ⟨“cs3 14784  Vtxcvtx 28976   WWalksNOn cwwlksnon 29807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-hash 14272  df-word 14455  df-concat 14512  df-s1 14537  df-s2 14790  df-s3 14791  df-wwlks 29810  df-wwlksn 29811  df-wwlksnon 29812

This theorem is referenced by:  elwwlks2on  29939  frgr2wwlk1  30308