Theorem wlkp1lem6 26986

Description: Lemma for wlkp1 26989. (Contributed by AV, 6-Mar-2021.)

Hypotheses

Ref	Expression
wlkp1.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkp1.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkp1.f	⊢ (𝜑 → Fun 𝐼)
wlkp1.a	⊢ (𝜑 → 𝐼 ∈ Fin)
wlkp1.b	⊢ (𝜑 → 𝐵 ∈ V)
wlkp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
wlkp1.d	⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w	⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
wlkp1.n	⊢ 𝑁 = (♯‘𝐹)
wlkp1.e	⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
wlkp1.x	⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
wlkp1.u	⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
wlkp1.h	⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
wlkp1.q	⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
wlkp1.s	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)

Assertion

Ref	Expression
wlkp1lem6	⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))

Distinct variable group:   𝜑,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)   𝑃(𝑘)   𝑄(𝑘)   𝑆(𝑘)   𝐸(𝑘)   𝐹(𝑘)   𝐺(𝑘)   𝐻(𝑘)   𝐼(𝑘)   𝑁(𝑘)   𝑉(𝑘)

Proof of Theorem wlkp1lem6

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkp1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		wlkp1.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3		wlkp1.f	. . . 4 ⊢ (𝜑 → Fun 𝐼)
4		wlkp1.a	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
5		wlkp1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
6		wlkp1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
7		wlkp1.d	. . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
8		wlkp1.w	. . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
9		wlkp1.n	. . . 4 ⊢ 𝑁 = (♯‘𝐹)
10		wlkp1.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
11		wlkp1.x	. . . 4 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
12		wlkp1.u	. . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
13		wlkp1.h	. . . 4 ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
14		wlkp1.q	. . . 4 ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
15		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem5 26985	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥))
17		elfzofz 12787	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (0...𝑁))
18	17	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (0...𝑁))
19		fveq2 6437	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑄‘𝑥) = (𝑄‘𝑘))
20		fveq2 6437	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑃‘𝑥) = (𝑃‘𝑘))
21	19, 20	eqeq12d 2840	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘𝑘) = (𝑃‘𝑘)))
22	21	rspcv 3522	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘𝑘) = (𝑃‘𝑘)))
23	18, 22	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘𝑘) = (𝑃‘𝑘)))
24	23	imp 397	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → (𝑄‘𝑘) = (𝑃‘𝑘))
25		fzofzp1 12867	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 + 1) ∈ (0...𝑁))
26	25	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ (0...𝑁))
27		fveq2 6437	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑘 + 1)))
28		fveq2 6437	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑘 + 1)))
29	27, 28	eqeq12d 2840	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
30	29	rspcv 3522	. . . . . 6 ⊢ ((𝑘 + 1) ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
31	26, 30	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
32	31	imp 397	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
33	12	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
34	13	fveq1i 6438	. . . . . . . 8 ⊢ (𝐻‘𝑘) = ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘)
35		fzonel 12785	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
36		eleq1 2894	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 𝑘 → (𝑁 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^𝑁)))
37	35, 36	mtbii 318	. . . . . . . . . . . . 13 ⊢ (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁))
38	37	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁)))
39	38	con2d 132	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑘))
40	39	imp 397	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ¬ 𝑁 = 𝑘)
41	40	neqned 3006	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ≠ 𝑘)
42		fvunsn 6702	. . . . . . . . 9 ⊢ (𝑁 ≠ 𝑘 → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹‘𝑘))
43	41, 42	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹‘𝑘))
44	34, 43	syl5eq 2873	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐻‘𝑘) = (𝐹‘𝑘))
45	33, 44	fveq12d 6444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)))
46	9	oveq2i 6921	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^(♯‘𝐹))
47	46	eleq2i 2898	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^(♯‘𝐹)))
48	2	wlkf 26919	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
49	8, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
50		wrdsymbcl 13594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom 𝐼)
51	50	ex 403	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐹‘𝑘) ∈ dom 𝐼))
52	49, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐹‘𝑘) ∈ dom 𝐼))
53	47, 52	syl5bi 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝐹‘𝑘) ∈ dom 𝐼))
54	53	imp 397	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐹‘𝑘) ∈ dom 𝐼)
55		eleq1 2894	. . . . . . . . . . . . 13 ⊢ (𝐵 = (𝐹‘𝑘) → (𝐵 ∈ dom 𝐼 ↔ (𝐹‘𝑘) ∈ dom 𝐼))
56	54, 55	syl5ibrcom 239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 = (𝐹‘𝑘) → 𝐵 ∈ dom 𝐼))
57	56	con3d 150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹‘𝑘)))
58	57	ex 403	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹‘𝑘))))
59	7, 58	mpid 44	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝐵 = (𝐹‘𝑘)))
60	59	imp 397	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ¬ 𝐵 = (𝐹‘𝑘))
61	60	neqned 3006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐵 ≠ (𝐹‘𝑘))
62		fvunsn 6702	. . . . . . 7 ⊢ (𝐵 ≠ (𝐹‘𝑘) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
63	61, 62	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
64	45, 63	eqtrd 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
65	64	adantr 474	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
66	24, 32, 65	3jca 1162	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → ((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))
67	16, 66	mpidan 680	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))
68	67	ralrimiva 3175	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  Vcvv 3414   ∪ cun 3796   ⊆ wss 3798  {csn 4399  {cpr 4401  ⟨cop 4405   class class class wbr 4875  dom cdm 5346  Fun wfun 6121  ‘cfv 6127  (class class class)co 6910  Fincfn 8228  0cc0 10259  1c1 10260   + caddc 10262  ...cfz 12626  ..^cfzo 12767  ♯chash 13417  Word cword 13581  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352  Walkscwlks 26901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-wlks 26904

This theorem is referenced by:  wlkp1lem8  26988