Theorem wlkp1lem6 29505

Description: Lemma for wlkp1 29508. (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 ∈ 𝑊)
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)   𝑊(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 ∈ 𝑊)
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 29504	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥))
17		elfzofz 13681	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (0...𝑁))
18	17	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (0...𝑁))
19		fveq2 6897	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑄‘𝑥) = (𝑄‘𝑘))
20		fveq2 6897	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑃‘𝑥) = (𝑃‘𝑘))
21	19, 20	eqeq12d 2744	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘𝑘) = (𝑃‘𝑘)))
22	21	rspcv 3605	. . . . . 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 406	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → (𝑄‘𝑘) = (𝑃‘𝑘))
25		fzofzp1 13762	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 + 1) ∈ (0...𝑁))
26	25	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ (0...𝑁))
27		fveq2 6897	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑘 + 1)))
28		fveq2 6897	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑘 + 1)))
29	27, 28	eqeq12d 2744	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
30	29	rspcv 3605	. . . . . 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 406	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
33	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
34	13	fveq1i 6898	. . . . . . . 8 ⊢ (𝐻‘𝑘) = ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘)
35		fzonel 13679	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
36		eleq1 2817	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 𝑘 → (𝑁 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^𝑁)))
37	35, 36	mtbii 326	. . . . . . . . . . . . 13 ⊢ (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁))
38	37	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁)))
39	38	con2d 134	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑘))
40	39	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ¬ 𝑁 = 𝑘)
41	40	neqned 2944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ≠ 𝑘)
42		fvunsn 7188	. . . . . . . . 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	eqtrid 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐻‘𝑘) = (𝐹‘𝑘))
45	33, 44	fveq12d 6904	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)))
46	9	oveq2i 7431	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^(♯‘𝐹))
47	46	eleq2i 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^(♯‘𝐹)))
48	2	wlkf 29441	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
49	8, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
50		wrdsymbcl 14510	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom 𝐼)
51	50	ex 412	. . . . . . . . . . . . . . . 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	biimtrid 241	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝐹‘𝑘) ∈ dom 𝐼))
54	53	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐹‘𝑘) ∈ dom 𝐼)
55		eleq1 2817	. . . . . . . . . . . . 13 ⊢ (𝐵 = (𝐹‘𝑘) → (𝐵 ∈ dom 𝐼 ↔ (𝐹‘𝑘) ∈ dom 𝐼))
56	54, 55	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 = (𝐹‘𝑘) → 𝐵 ∈ dom 𝐼))
57	56	con3d 152	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹‘𝑘)))
58	57	ex 412	. . . . . . . . . 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 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ¬ 𝐵 = (𝐹‘𝑘))
61	60	neqned 2944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐵 ≠ (𝐹‘𝑘))
62		fvunsn 7188	. . . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
65	64	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
66	24, 32, 65	3jca 1126	. . 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 688	. 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 3143	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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  ∀wral 3058   ∪ cun 3945   ⊆ wss 3947  {csn 4629  {cpr 4631  ⟨cop 4635   class class class wbr 5148  dom cdm 5678  Fun wfun 6542  ‘cfv 6548  (class class class)co 7420  Fincfn 8964  0cc0 11139  1c1 11140   + caddc 11142  ...cfz 13517  ..^cfzo 13660  ♯chash 14322  Word cword 14497  Vtxcvtx 28822  iEdgciedg 28823  Edgcedg 28873  Walkscwlks 29423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-fzo 13661  df-hash 14323  df-word 14498  df-wlks 29426

This theorem is referenced by:  wlkp1lem8  29507