Theorem cycl3grtrilem 48300

Description: Lemma for cycl3grtri 48301. (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 29810	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
2		eqid 2737	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	upgrwlkvtxedg 29730	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑥 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺))
4	1, 3	sylan2 594	. . 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 7376	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (0..^(♯‘𝐹)) = (0..^3))
7		fzo0to3tp 13680	. . . . . . 7 ⊢ (0..^3) = {0, 1, 2}
8	6, 7	eqtrdi 2788	. . . . . 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 3298	. . 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 6842	. . . . . . 7 ⊢ ((♯‘𝐹) = 3 → (𝑃‘(♯‘𝐹)) = (𝑃‘3))
13	12	eqeq2d 2748	. . . . . 6 ⊢ ((♯‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
14		c0ex 11138	. . . . . . . 8 ⊢ 0 ∈ V
15		1ex 11140	. . . . . . . 8 ⊢ 1 ∈ V
16		2ex 12234	. . . . . . . 8 ⊢ 2 ∈ V
17		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
18		fv0p1e1 12275	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑃‘(𝑥 + 1)) = (𝑃‘1))
19	17, 18	preq12d 4700	. . . . . . . . 9 ⊢ (𝑥 = 0 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘0), (𝑃‘1)})
20	19	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 0 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
21		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘𝑥) = (𝑃‘1))
22		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
23		1p1e2 12277	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
24	22, 23	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 + 1) = 2)
25	24	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝑃‘(𝑥 + 1)) = (𝑃‘2))
26	21, 25	preq12d 4700	. . . . . . . . 9 ⊢ (𝑥 = 1 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘1), (𝑃‘2)})
27	26	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 1 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
28		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘𝑥) = (𝑃‘2))
29		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑥 = 2 → (𝑥 + 1) = (2 + 1))
30		2p1e3 12294	. . . . . . . . . . . 12 ⊢ (2 + 1) = 3
31	29, 30	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑥 = 2 → (𝑥 + 1) = 3)
32	31	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑥 = 2 → (𝑃‘(𝑥 + 1)) = (𝑃‘3))
33	28, 32	preq12d 4700	. . . . . . . . 9 ⊢ (𝑥 = 2 → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑃‘2), (𝑃‘3)})
34	33	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 2 → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘2), (𝑃‘3)} ∈ (Edg‘𝐺)))
35	14, 15, 16, 20, 27, 34	raltp 4664	. . . . . . 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 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‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
37		preq2 4693	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘0)} = {(𝑃‘2), (𝑃‘3)})
38		prcom 4691	. . . . . . . . . . . . . 14 ⊢ {(𝑃‘2), (𝑃‘0)} = {(𝑃‘0), (𝑃‘2)}
39	37, 38	eqtr3di 2787	. . . . . . . . . . . . 13 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘0), (𝑃‘2)})
40	39	eleq1d 2822	. . . . . . . . . . . 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 1136	. . . . . . . . . 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 1197	. . . . . . . . 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 1129	. . . . . . . 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {cpr 4584  {ctp 4586   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  0cc0 11038  1c1 11039   + caddc 11041  2c2 12212  3c3 12213  ..^cfzo 13582  ♯chash 14265  Edgcedg 29132  UPGraphcupgr 29165  Walkscwlks 29682  Pathscpths 29795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-hash 14266  df-word 14449  df-edg 29133  df-uhgr 29143  df-upgr 29167  df-wlks 29685  df-trls 29776  df-pths 29799

This theorem is referenced by:  cycl3grtri  48301