Theorem cycl3grtrilem 49259

Description: Lemma for cycl3grtri 49260. (Contributed by AV, 5-Oct-2025.)

Assertion

Ref	Expression
cycl3grtrilem	⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))

Proof of Theorem cycl3grtrilem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pthiswlk 30343	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
2		eqid 2763	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	upgrwlkvtxedg 30258	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺))
4	1, 3	sylan2 605	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) → ∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺))
5	4	adantr 486	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺))
6		oveq2 7454	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = (0..^3))
7		fzo0to3tp 13917	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
8	6, 7	eqtrdi 2814	. . . . . 6 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = {0, 1, 2})
9	8	adantl 487	. . . . 5 ⊢ (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (0..^(♯‘𝐹)) = {0, 1, 2})
10	9	adantl 487	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (0..^(♯‘𝐹)) = {0, 1, 2})
11	10	raleqdv 3323	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑥 ∈ {0, 1, 2} {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺)))
12		fveq2 6911	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (𝑃‘(♯‘𝐹)) = (𝑃‘3))
13	12	eqeq2d 2774	. . . . . 6 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
14		c0ex 11330	. . . . . . . 8 ⊢ 0 ∈ V
15		1ex 11333	. . . . . . . 8 ⊢ 1 ∈ V
16		2ex 12450	. . . . . . . 8 ⊢ 2 ∈ V
17		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
18		fv0p1e1 12494	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘(𝑥 + 1)) = (𝑃‘1))
19	17, 18	preq12d 4702	. . . . . . . . 9 ⊢ (𝑥 = 0 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘0), (𝑃‘1)})
20	19	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 0 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
21		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘𝑥) = (𝑃‘1))
22		oveq1 7453	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
23		1p1e2 12496	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
24	22, 23	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 + 1) = 2)
25	24	fveq2d 6915	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘(𝑥 + 1)) = (𝑃‘2))
26	21, 25	preq12d 4702	. . . . . . . . 9 ⊢ (𝑥 = 1 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘1), (𝑃‘2)})
27	26	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 1 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
28		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘𝑥) = (𝑃‘2))
29		oveq1 7453	. . . . . . . . . . . 12 ⊢ (𝑥 = 2 → (𝑥 + 1) = (2 + 1))
30		2p1e3 12514	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
31	29, 30	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑥 = 2 → (𝑥 + 1) = 3)
32	31	fveq2d 6915	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘(𝑥 + 1)) = (𝑃‘3))
33	28, 32	preq12d 4702	. . . . . . . . 9 ⊢ (𝑥 = 2 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘2), (𝑃‘3)})
34	33	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 2 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺)))
35	14, 15, 16, 20, 27, 34	raltp 4666	. . . . . . 7 ⊢ (∀𝑥 ∈ {0, 1, 2} {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺)))
36		simpr1 1213	. . . . . . . . 9 ⊢ (((𝑃‘0) = (𝑃‘3) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺))) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
37		preq2 4695	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘0)} = {(𝑃‘2), (𝑃‘3)})
38		prcom 4693	. . . . . . . . . . . . . 14 ⊢ {(𝑃‘2), (𝑃‘0)} = {(𝑃‘0), (𝑃‘2)}
39	37, 38	eqtr3di 2813	. . . . . . . . . . . . 13 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘0), (𝑃‘2)})
40	39	eleq1d 2849	. . . . . . . . . . . 12 ⊢ ((𝑃‘0) = (𝑃‘3) → ({(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺)))
41	40	biimpcd 252	. . . . . . . . . . 11 ⊢ ({(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺) → ((𝑃‘0) = (𝑃‘3) → {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺)))
42	41	3ad2ant3 1153	. . . . . . . . . 10 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺)) → ((𝑃‘0) = (𝑃‘3) → {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺)))
43	42	impcom 413	. . . . . . . . 9 ⊢ (((𝑃‘0) = (𝑃‘3) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺))) → {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺))
44		simpr2 1214	. . . . . . . . 9 ⊢ (((𝑃‘0) = (𝑃‘3) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺))) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
45	36, 43, 44	3jca 1146	. . . . . . . 8 ⊢ (((𝑃‘0) = (𝑃‘3) ∧ ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺))) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
46	45	ex 418	. . . . . . 7 ⊢ ((𝑃‘0) = (𝑃‘3) → (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺)) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
47	35, 46	biimtrid 245	. . . . . 6 ⊢ ((𝑃‘0) = (𝑃‘3) → (∀𝑥 ∈ {0, 1, 2} {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
48	13, 47	biimtrdi 256	. . . . 5 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (∀𝑥 ∈ {0, 1, 2} {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))))
49	48	impcom 413	. . . 4 ⊢ (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (∀𝑥 ∈ {0, 1, 2} {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
50	49	adantl 487	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (∀𝑥 ∈ {0, 1, 2} {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
51	11, 50	sylbid 243	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))))
52	5, 51	mpd 16	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {cpr 4590  {ctp 4592   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  0cc0 11230  1c1 11231   + caddc 11233  2c2 12427  3c3 12428  ..^cfzo 13818  ♯chash 14504  Edgcedg 29658  UPGraphcupgr 29691  Walkscwlks 30210  Pathscpths 30328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8748  df-map 8880  df-pm 8881  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-dju 10012  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-n0 12637  df-xnn0 12710  df-z 12724  df-uz 12996  df-fz 13670  df-fzo 13819  df-hash 14505  df-word 14689  df-edg 29659  df-uhgr 29669  df-upgr 29693  df-wlks 30213  df-trls 30308  df-pths 30332

This theorem is used by:  cycl3grtri  49260