Theorem cycl3grtrilem 47900

Description: Lemma for cycl3grtri 47901. (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 29662	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
2		eqid 2730	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	upgrwlkvtxedg 29580	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺))
4	1, 3	sylan2 593	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) → ∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺))
5	4	adantr 480	. 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 7402	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = (0..^3))
7		fzo0to3tp 13725	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
8	6, 7	eqtrdi 2781	. . . . . 6 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = {0, 1, 2})
9	8	adantl 481	. . . . 5 ⊢ (((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3) → (0..^(♯‘𝐹)) = {0, 1, 2})
10	9	adantl 481	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃) ∧ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ∧ (♯‘𝐹) = 3)) → (0..^(♯‘𝐹)) = {0, 1, 2})
11	10	raleqdv 3302	. . 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 6865	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (𝑃‘(♯‘𝐹)) = (𝑃‘3))
13	12	eqeq2d 2741	. . . . . 6 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
14		c0ex 11186	. . . . . . . 8 ⊢ 0 ∈ V
15		1ex 11188	. . . . . . . 8 ⊢ 1 ∈ V
16		2ex 12274	. . . . . . . 8 ⊢ 2 ∈ V
17		fveq2 6865	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
18		fv0p1e1 12320	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘(𝑥 + 1)) = (𝑃‘1))
19	17, 18	preq12d 4713	. . . . . . . . 9 ⊢ (𝑥 = 0 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘0), (𝑃‘1)})
20	19	eleq1d 2814	. . . . . . . 8 ⊢ (𝑥 = 0 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
21		fveq2 6865	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘𝑥) = (𝑃‘1))
22		oveq1 7401	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
23		1p1e2 12322	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
24	22, 23	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 + 1) = 2)
25	24	fveq2d 6869	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘(𝑥 + 1)) = (𝑃‘2))
26	21, 25	preq12d 4713	. . . . . . . . 9 ⊢ (𝑥 = 1 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘1), (𝑃‘2)})
27	26	eleq1d 2814	. . . . . . . 8 ⊢ (𝑥 = 1 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
28		fveq2 6865	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘𝑥) = (𝑃‘2))
29		oveq1 7401	. . . . . . . . . . . 12 ⊢ (𝑥 = 2 → (𝑥 + 1) = (2 + 1))
30		2p1e3 12339	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
31	29, 30	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (𝑥 = 2 → (𝑥 + 1) = 3)
32	31	fveq2d 6869	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘(𝑥 + 1)) = (𝑃‘3))
33	28, 32	preq12d 4713	. . . . . . . . 9 ⊢ (𝑥 = 2 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘2), (𝑃‘3)})
34	33	eleq1d 2814	. . . . . . . 8 ⊢ (𝑥 = 2 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺)))
35	14, 15, 16, 20, 27, 34	raltp 4677	. . . . . . 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 1195	. . . . . . . . 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 4706	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘0)} = {(𝑃‘2), (𝑃‘3)})
38		prcom 4704	. . . . . . . . . . . . . 14 ⊢ {(𝑃‘2), (𝑃‘0)} = {(𝑃‘0), (𝑃‘2)}
39	37, 38	eqtr3di 2780	. . . . . . . . . . . . 13 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘0), (𝑃‘2)})
40	39	eleq1d 2814	. . . . . . . . . . . 12 ⊢ ((𝑃‘0) = (𝑃‘3) → ({(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺)))
41	40	biimpcd 249	. . . . . . . . . . 11 ⊢ ({(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺) → ((𝑃‘0) = (𝑃‘3) → {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺)))
42	41	3ad2ant3 1135	. . . . . . . . . 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 407	. . . . . . . . 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 1196	. . . . . . . . 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 1128	. . . . . . . 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 412	. . . . . . 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 242	. . . . . 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 253	. . . . 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 407	. . . 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 481	. . 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 240	. 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 15	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)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3046  {cpr 4599  {ctp 4601   class class class wbr 5115  ‘cfv 6519  (class class class)co 7394  0cc0 11086  1c1 11087   + caddc 11089  2c2 12252  3c3 12253  ..^cfzo 13628  ♯chash 14305  Edgcedg 28981  UPGraphcupgr 29014  Walkscwlks 29531  Pathscpths 29647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718  ax-cnex 11142  ax-resscn 11143  ax-1cn 11144  ax-icn 11145  ax-addcl 11146  ax-addrcl 11147  ax-mulcl 11148  ax-mulrcl 11149  ax-mulcom 11150  ax-addass 11151  ax-mulass 11152  ax-distr 11153  ax-i2m1 11154  ax-1ne0 11155  ax-1rid 11156  ax-rnegex 11157  ax-rrecex 11158  ax-cnre 11159  ax-pre-lttri 11160  ax-pre-lttrn 11161  ax-pre-ltadd 11162  ax-pre-mulgt0 11163

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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-nel 3032  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-tp 4602  df-op 4604  df-uni 4880  df-int 4919  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-1st 7977  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-1o 8443  df-2o 8444  df-oadd 8447  df-er 8682  df-map 8805  df-pm 8806  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-dju 9872  df-card 9910  df-pnf 11228  df-mnf 11229  df-xr 11230  df-ltxr 11231  df-le 11232  df-sub 11425  df-neg 11426  df-nn 12198  df-2 12260  df-3 12261  df-n0 12459  df-xnn0 12532  df-z 12546  df-uz 12810  df-fz 13482  df-fzo 13629  df-hash 14306  df-word 14489  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-wlks 29534  df-trls 29627  df-pths 29651

This theorem is referenced by:  cycl3grtri  47901