Theorem umgr2adedgwlkonALT 29745

Description: Alternate proof for umgr2adedgwlkon 29744, using umgr2adedgwlk 29743, 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 29743	. . 3 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
10		simp1 1134	. . . 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 2735	. . . . . 6 ⊢ (𝐴 = (𝑃‘0) → (𝑃‘0) = 𝐴)
13	12	3ad2ant1 1131	. . . . 5 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘0) = 𝐴)
14	13	3ad2ant3 1133	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘0) = 𝐴)
15		fveq2 6891	. . . . . . . . . . . 12 ⊢ (2 = (♯‘𝐹) → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
16	15	eqcoms 2735	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 2 → (𝑃‘2) = (𝑃‘(♯‘𝐹)))
17	16	eqeq1d 2729	. . . . . . . . . 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 2735	. . . . . . . 8 ⊢ (𝐶 = (𝑃‘2) → ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = 𝐶))
20	19	3ad2ant3 1133	. . . . . . 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 1109	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘(♯‘𝐹)) = 𝐶)
24	10, 14, 23	3jca 1126	. . 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 1093	. . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
27	5, 6, 26	sylanbrc 582	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
28	1	umgr2adedgwlklem 29742	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
29		3simpb 1147	. . . . . 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 1093	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) ↔ (𝐺 ∈ UMGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
33	5, 31, 32	sylanbrc 582	. . 3 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
34		s2cli 14855	. . . . 5 ⊢ ⟨“𝐽𝐾”⟩ ∈ Word V
35	3, 34	eqeltri 2824	. . . 4 ⊢ 𝐹 ∈ Word V
36		s3cli 14856	. . . . 5 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
37	4, 36	eqeltri 2824	. . . 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 1128	. . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))
41	40	anim1i 614	. . . 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 2727	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
43	42	iswlkon 29458	. . . 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 585	. 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  Vcvv 3469  {cpr 4626   class class class wbr 5142  ‘cfv 6542  (class class class)co 7414  0cc0 11130  1c1 11131  2c2 12289  ♯chash 14313  Word cword 14488  ⟨“cs2 14816  ⟨“cs3 14817  Vtxcvtx 28796  iEdgciedg 28797  Edgcedg 28847  UMGraphcumgr 28881  Walkscwlks 29397  WalksOncwlkson 29398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-s2 14823  df-s3 14824  df-edg 28848  df-umgr 28883  df-wlks 29400  df-wlkson 29401

This theorem is referenced by: (None)