Theorem clwlkclwwlkflem 30321

Description: Lemma for clwlkclwwlkf 30325. (Contributed by AV, 24-May-2022.)

Hypotheses

Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.a	⊢ 𝐴 = (1st ‘𝑈)
clwlkclwwlkf.b	⊢ 𝐵 = (2nd ‘𝑈)

Assertion

Ref	Expression
clwlkclwwlkflem	⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))

Distinct variable groups:   𝑤,𝐺   𝑤,𝐴   𝑤,𝑈

Allowed substitution hints:   𝐵(𝑤)   𝐶(𝑤)

Proof of Theorem clwlkclwwlkflem

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . 6 ⊢ (𝑤 = 𝑈 → (1st ‘𝑤) = (1st ‘𝑈))
2		clwlkclwwlkf.a	. . . . . 6 ⊢ 𝐴 = (1st ‘𝑈)
3	1, 2	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑈 → (1st ‘𝑤) = 𝐴)
4	3	fveq2d 6885	. . . 4 ⊢ (𝑤 = 𝑈 → (♯‘(1st ‘𝑤)) = (♯‘𝐴))
5	4	breq2d 5120	. . 3 ⊢ (𝑤 = 𝑈 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘𝐴)))
6		clwlkclwwlkf.c	. . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
7	5, 6	elrab2 3653	. 2 ⊢ (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8		clwlkwlk 30090	. . . 4 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9		wlkop 29943	. . . . 5 ⊢ (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
10		clwlkclwwlkf.b	. . . . . . . 8 ⊢ 𝐵 = (2nd ‘𝑈)
11	2, 10	opeq12i 4842	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
12	11	eqeq2i 2774	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
13		eleq1 2849	. . . . . . 7 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺)))
14		df-br 5109	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺))
15		isclwlk 30088	. . . . . . . . 9 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))))
16		wlkcl 29931	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐴) ∈ ℕ0)
17		elnnnn0c 12548	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴)))
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴))))
19	16, 18	mpbirand 719	. . . . . . . . . . . . . 14 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ 1 ≤ (♯‘𝐴)))
20	19	bicomd 226	. . . . . . . . . . . . 13 ⊢ (𝐴(Walks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
21	20	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
22	21	pm5.32i 584	. . . . . . . . . . 11 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
23		df-3an 1103	. . . . . . . . . . 11 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
24	22, 23	sylbb2 241	. . . . . . . . . 10 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
25	24	ex 417	. . . . . . . . 9 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
26	15, 25	sylbi 220	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
27	14, 26	sylbir 238	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
28	13, 27	biimtrdi 256	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
29	12, 28	sylbir 238	. . . . 5 ⊢ (𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
30	9, 29	syl 18	. . . 4 ⊢ (𝑈 ∈ (Walks‘𝐺) → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
31	8, 30	mpcom 39	. . 3 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
32	31	imp 411	. 2 ⊢ ((𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
33	7, 32	sylbi 220	1 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {crab 3414  ⟨cop 4594   class class class wbr 5108  ‘cfv 6536  1^st c1st 7983  2^nd c2nd 7984  0cc0 11099  1c1 11100   ≤ cle 11243  ℕcn 12232  ℕ₀cn0 12503  ♯chash 14365  Walkscwlks 29912  ClWalkscclwlks 30085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  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 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13682  df-hash 14366  df-word 14550  df-wlks 29915  df-clwlks 30086

This theorem is referenced by:  clwlkclwwlkf1lem2  30322  clwlkclwwlkf1lem3  30323  clwlkclwwlkf  30325  clwlkclwwlkf1  30327