Theorem umgr2adedgwlkonALT 29814

Description: Alternate proof for umgr2adedgwlkon 29813, using umgr2adedgwlk 29812, 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 29812	. . 3 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
10		simp1 1133	. . . 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 2733	. . . . . 6 ⊢ (𝐴 = (𝑃‘0) → (𝑃‘0) = 𝐴)
13	12	3ad2ant1 1130	. . . . 5 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘0) = 𝐴)
14	13	3ad2ant3 1132	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘0) = 𝐴)
15		fveq2 6894	. . . . . . . . . . . 12 ⊢ (2 = (♯‘𝐹) → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
16	15	eqcoms 2733	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
17	16	eqeq1d 2727	. . . . . . . . . 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 2733	. . . . . . . 8 ⊢ (𝐶 = (𝑃‘2) → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
20	19	3ad2ant3 1132	. . . . . . 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 1108	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘(♯‘𝐹)) = 𝐶)
24	10, 14, 23	3jca 1125	. . 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 1092	. . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
27	5, 6, 26	sylanbrc 581	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
28	1	umgr2adedgwlklem 29811	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
29		3simpb 1146	. . . . . 6 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
30	29	adantl 480	. . . . 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 1092	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ↔ (𝐺 ∈ UMGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
33	5, 31, 32	sylanbrc 581	. . 3 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
34		s2cli 14863	. . . . 5 ⊢ ⟨“𝐽𝐾”⟩ ∈ Word V
35	3, 34	eqeltri 2821	. . . 4 ⊢ 𝐹 ∈ Word V
36		s3cli 14864	. . . . 5 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
37	4, 36	eqeltri 2821	. . . 4 ⊢ 𝑃 ∈ Word V
38	35, 37	pm3.2i 469	. . 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 1127	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
41	40	anim1i 613	. . . 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 2725	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
43	42	iswlkon 29527	. . . 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 584	. 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  Vcvv 3463  {cpr 4631   class class class wbr 5148  ‘cfv 6547  (class class class)co 7417  0cc0 11138  1c1 11139  2c2 12297  ♯chash 14321  Word cword 14496  ⟨“cs2 14824  ⟨“cs3 14825  Vtxcvtx 28865  iEdgciedg 28866  Edgcedg 28916  UMGraphcumgr 28950  Walkscwlks 29466  WalksOncwlkson 29467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-hash 14322  df-word 14497  df-concat 14553  df-s1 14578  df-s2 14831  df-s3 14832  df-edg 28917  df-umgr 28952  df-wlks 29469  df-wlkson 29470

This theorem is referenced by: (None)