Theorem umgr2adedgwlkonALT 29190

Description: Alternate proof for umgr2adedgwlkon 29189, using umgr2adedgwlk 29188, but with a much longer proof! In a multigraph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
umgr2adedgwlk.e	⊢ 𝐸 = (Edg‘𝐺)
umgr2adedgwlk.i	⊢ 𝐼 = (iEdg‘𝐺)
umgr2adedgwlk.f	⊢ 𝐹 = ⟨“𝐽𝐾”⟩
umgr2adedgwlk.p	⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩
umgr2adedgwlk.g	⊢ (𝜑 → 𝐺 ∈ UMGraph)
umgr2adedgwlk.a	⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
umgr2adedgwlk.j	⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵})
umgr2adedgwlk.k	⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶})

Assertion

Ref	Expression
umgr2adedgwlkonALT	⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Proof of Theorem umgr2adedgwlkonALT

Step	Hyp	Ref	Expression
1		umgr2adedgwlk.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
2		umgr2adedgwlk.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3		umgr2adedgwlk.f	. . . 4 ⊢ 𝐹 = ⟨“𝐽𝐾”⟩
4		umgr2adedgwlk.p	. . . 4 ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩
5		umgr2adedgwlk.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ UMGraph)
6		umgr2adedgwlk.a	. . . 4 ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
7		umgr2adedgwlk.j	. . . 4 ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵})
8		umgr2adedgwlk.k	. . . 4 ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶})
9	1, 2, 3, 4, 5, 6, 7, 8	umgr2adedgwlk 29188	. . 3 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
10		simp1 1136	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → 𝐹(Walks‘𝐺)𝑃)
11		id 22	. . . . . . 7 ⊢ ((𝑃‘0) = 𝐴 → (𝑃‘0) = 𝐴)
12	11	eqcoms 2740	. . . . . 6 ⊢ (𝐴 = (𝑃‘0) → (𝑃‘0) = 𝐴)
13	12	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘0) = 𝐴)
14	13	3ad2ant3 1135	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘0) = 𝐴)
15		fveq2 6888	. . . . . . . . . . . 12 ⊢ (2 = (♯‘𝐹) → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
16	15	eqcoms 2740	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
17	16	eqeq1d 2734	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘2) = 𝐶 ↔ (𝑃‘(♯‘𝐹)) = 𝐶))
18	17	biimpcd 248	. . . . . . . . 9 ⊢ ((𝑃‘2) = 𝐶 → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
19	18	eqcoms 2740	. . . . . . . 8 ⊢ (𝐶 = (𝑃‘2) → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
20	19	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
21	20	com12 32	. . . . . 6 ⊢ ((♯‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘(♯‘𝐹)) = 𝐶))
22	21	a1i 11	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘(♯‘𝐹)) = 𝐶)))
23	22	3imp 1111	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘(♯‘𝐹)) = 𝐶)
24	10, 14, 23	3jca 1128	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))
25	9, 24	syl 17	. 2 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))
26		3anass 1095	. . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
27	5, 6, 26	sylanbrc 583	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
28	1	umgr2adedgwlklem 29187	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
29		3simpb 1149	. . . . . 6 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
30	29	adantl 482	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
31	27, 28, 30	3syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
32		3anass 1095	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ↔ (𝐺 ∈ UMGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
33	5, 31, 32	sylanbrc 583	. . 3 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
34		s2cli 14827	. . . . 5 ⊢ ⟨“𝐽𝐾”⟩ ∈ Word V
35	3, 34	eqeltri 2829	. . . 4 ⊢ 𝐹 ∈ Word V
36		s3cli 14828	. . . . 5 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
37	4, 36	eqeltri 2829	. . . 4 ⊢ 𝑃 ∈ Word V
38	35, 37	pm3.2i 471	. . 3 ⊢ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)
39		id 22	. . . . . 6 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
40	39	3adant1 1130	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
41	40	anim1i 615	. . . 4 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)))
42		eqid 2732	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
43	42	iswlkon 28903	. . . 4 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶)))
44	41, 43	syl 17	. . 3 ⊢ (((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶)))
45	33, 38, 44	sylancl 586	. 2 ⊢ (𝜑 → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶)))
46	25, 45	mpbird 256	1 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474  {cpr 4629   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107  2c2 12263  ♯chash 14286  Word cword 14460  ⟨“cs2 14788  ⟨“cs3 14789  Vtxcvtx 28245  iEdgciedg 28246  Edgcedg 28296  UMGraphcumgr 28330  Walkscwlks 28842  WalksOncwlkson 28843

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-s2 14795  df-s3 14796  df-edg 28297  df-umgr 28332  df-wlks 28845  df-wlkson 28846

This theorem is referenced by: (None)