Theorem cycl3grtrilem 48988

Description: Lemma for cycl3grtri 48989. (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 30292	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
2		eqid 2761	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	upgrwlkvtxedg 30207	. . . 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 7420	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = (0..^3))
7		fzo0to3tp 13867	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
8	6, 7	eqtrdi 2812	. . . . . 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 3320	. . 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 6877	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (𝑃‘(♯‘𝐹)) = (𝑃‘3))
13	12	eqeq2d 2772	. . . . . 6 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
14		c0ex 11281	. . . . . . . 8 ⊢ 0 ∈ V
15		1ex 11284	. . . . . . . 8 ⊢ 1 ∈ V
16		2ex 12401	. . . . . . . 8 ⊢ 2 ∈ V
17		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
18		fv0p1e1 12445	. . . . . . . . . 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 2846	. . . . . . . 8 ⊢ (𝑥 = 0 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
21		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘𝑥) = (𝑃‘1))
22		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
23		1p1e2 12447	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
24	22, 23	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 + 1) = 2)
25	24	fveq2d 6881	. . . . . . . . . 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 2846	. . . . . . . 8 ⊢ (𝑥 = 1 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
28		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘𝑥) = (𝑃‘2))
29		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑥 = 2 → (𝑥 + 1) = (2 + 1))
30		2p1e3 12465	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
31	29, 30	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑥 = 2 → (𝑥 + 1) = 3)
32	31	fveq2d 6881	. . . . . . . . . 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 2846	. . . . . . . 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 2811	. . . . . . . . . . . . 13 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘0), (𝑃‘2)})
40	39	eleq1d 2846	. . . . . . . . . . . 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 2145  ∀wral 3077  {cpr 4586  {ctp 4588   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  0cc0 11181  1c1 11182   + caddc 11184  2c2 12378  3c3 12379  ..^cfzo 13768  ♯chash 14454  Edgcedg 29607  UPGraphcupgr 29640  Walkscwlks 30159  Pathscpths 30277

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  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 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-oadd 8464  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9963  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-xnn0 12661  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-hash 14455  df-word 14639  df-edg 29608  df-uhgr 29618  df-upgr 29642  df-wlks 30162  df-trls 30257  df-pths 30281

This theorem is used by:  cycl3grtri  48989