Theorem umgr2adedgwlkonALT 29861

Description: Alternate proof for umgr2adedgwlkon 29860, using umgr2adedgwlk 29859, 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 29859	. . 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 2742	. . . . . 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 6872	. . . . . . . . . . . 12 ⊢ (2 = (♯‘𝐹) → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
16	15	eqcoms 2742	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
17	16	eqeq1d 2736	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘2) = 𝐶 ↔ (𝑃‘(♯‘𝐹)) = 𝐶))
18	17	biimpcd 249	. . . . . . . . 9 ⊢ ((𝑃‘2) = 𝐶 → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
19	18	eqcoms 2742	. . . . . . . 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 1110	. . . 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 1094	. . . . . 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 29858	. . . . 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 481	. . . . 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 1094	. . . 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 14886	. . . . 5 ⊢ ⟨“𝐽𝐾”⟩ ∈ Word V
35	3, 34	eqeltri 2829	. . . 4 ⊢ 𝐹 ∈ Word V
36		s3cli 14887	. . . . 5 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
37	4, 36	eqeltri 2829	. . . 4 ⊢ 𝑃 ∈ Word V
38	35, 37	pm3.2i 470	. . 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 2734	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
43	42	iswlkon 29569	. . . 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 257	1 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  Vcvv 3457  {cpr 4601   class class class wbr 5116  ‘cfv 6527  (class class class)co 7399  0cc0 11121  1c1 11122  2c2 12287  ♯chash 14336  Word cword 14519  ⟨“cs2 14847  ⟨“cs3 14848  Vtxcvtx 28907  iEdgciedg 28908  Edgcedg 28958  UMGraphcumgr 28992  Walkscwlks 29508  WalksOncwlkson 29509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-cnex 11177  ax-resscn 11178  ax-1cn 11179  ax-icn 11180  ax-addcl 11181  ax-addrcl 11182  ax-mulcl 11183  ax-mulrcl 11184  ax-mulcom 11185  ax-addass 11186  ax-mulass 11187  ax-distr 11188  ax-i2m1 11189  ax-1ne0 11190  ax-1rid 11191  ax-rnegex 11192  ax-rrecex 11193  ax-cnre 11194  ax-pre-lttri 11195  ax-pre-lttrn 11196  ax-pre-ltadd 11197  ax-pre-mulgt0 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-tp 4604  df-op 4606  df-uni 4881  df-int 4920  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-om 7856  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-1o 8474  df-oadd 8478  df-er 8713  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-dju 9907  df-card 9945  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11460  df-neg 11461  df-nn 12233  df-2 12295  df-3 12296  df-n0 12494  df-z 12581  df-uz 12845  df-fz 13514  df-fzo 13661  df-hash 14337  df-word 14520  df-concat 14576  df-s1 14601  df-s2 14854  df-s3 14855  df-edg 28959  df-umgr 28994  df-wlks 29511  df-wlkson 29512

This theorem is referenced by: (None)