Theorem wlkp1lem6 30239

Description: Lemma for wlkp1 30242. (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 30238	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥))
17		elfzofz 13790	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (0...𝑁))
18	17	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (0...𝑁))
19		fveq2 6877	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑄‘𝑥) = (𝑄‘𝑘))
20		fveq2 6877	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑃‘𝑥) = (𝑃‘𝑘))
21	19, 20	eqeq12d 2777	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘𝑘) = (𝑃‘𝑘)))
22	21	rspcv 3573	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘𝑘) = (𝑃‘𝑘)))
23	18, 22	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘𝑘) = (𝑃‘𝑘)))
24	23	imp 412	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → (𝑄‘𝑘) = (𝑃‘𝑘))
25		fzofzp1 13879	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑘 + 1) ∈ (0...𝑁))
26	25	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ (0...𝑁))
27		fveq2 6877	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑘 + 1)))
28		fveq2 6877	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑘 + 1)))
29	27, 28	eqeq12d 2777	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
30	29	rspcv 3573	. . . . . 6 ⊢ ((𝑘 + 1) ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
31	26, 30	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
32	31	imp 412	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
33	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
34	13	fveq1i 6878	. . . . . . . 8 ⊢ (𝐻‘𝑘) = ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘)
35		fzonel 13788	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
36		eleq1 2849	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 𝑘 → (𝑁 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^𝑁)))
37	35, 36	mtbii 329	. . . . . . . . . . . . 13 ⊢ (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁))
38	37	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁)))
39	38	con2d 135	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑘))
40	39	imp 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ¬ 𝑁 = 𝑘)
41	40	neqned 2963	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ≠ 𝑘)
42		fvunsn 7176	. . . . . . . . 9 ⊢ (𝑁 ≠ 𝑘 → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹‘𝑘))
43	41, 42	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹‘𝑘))
44	34, 43	eqtrid 2808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐻‘𝑘) = (𝐹‘𝑘))
45	33, 44	fveq12d 6884	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)))
46	9	oveq2i 7423	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) = (0..^(♯‘𝐹))
47	46	eleq2i 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^(♯‘𝐹)))
48	2	wlkf 30177	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
49	8, 48	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
50		wrdsymbcl 14652	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom 𝐼)
51	50	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐹‘𝑘) ∈ dom 𝐼))
52	49, 51	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐹‘𝑘) ∈ dom 𝐼))
53	47, 52	biimtrid 245	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝐹‘𝑘) ∈ dom 𝐼))
54	53	imp 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐹‘𝑘) ∈ dom 𝐼)
55		eleq1 2849	. . . . . . . . . . . . 13 ⊢ (𝐵 = (𝐹‘𝑘) → (𝐵 ∈ dom 𝐼 ↔ (𝐹‘𝑘) ∈ dom 𝐼))
56	54, 55	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 = (𝐹‘𝑘) → 𝐵 ∈ dom 𝐼))
57	56	con3d 153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹‘𝑘)))
58	57	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹‘𝑘))))
59	7, 58	mpid 45	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝐵 = (𝐹‘𝑘)))
60	59	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ¬ 𝐵 = (𝐹‘𝑘))
61	60	neqned 2963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐵 ≠ (𝐹‘𝑘))
62		fvunsn 7176	. . . . . . 7 ⊢ (𝐵 ≠ (𝐹‘𝑘) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
63	61, 62	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
64	45, 63	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
65	64	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥)) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
66	24, 32, 65	3jca 1146	. . 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 702	. 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 3155	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ∪ cun 3897   ⊆ wss 3899  {csn 4584  {cpr 4586  ⟨cop 4590   class class class wbr 5103  dom cdm 5651  Fun wfun 6525  ‘cfv 6531  (class class class)co 7412  Fincfn 8957  0cc0 11181  1c1 11182   + caddc 11184  ...cfz 13620  ..^cfzo 13768  ♯chash 14454  Word cword 14638  Vtxcvtx 29556  iEdgciedg 29557  Edgcedg 29607  Walkscwlks 30159

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-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-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  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-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-hash 14455  df-word 14639  df-wlks 30162

This theorem is used by:  wlkp1lem8  30241