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Theorem clwlkclwwlkflem 29525

Description: Lemma for clwlkclwwlkf 29529. (Contributed by AV, 24-May-2022.)

**Hypotheses**
Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
clwlkclwwlkf.a	⊢ 𝐴 = (1^st ‘𝑈)
clwlkclwwlkf.b	⊢ 𝐵 = (2^nd ‘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 6891	. . . . . 6 ⊢ (𝑤 = 𝑈 → (1^st ‘𝑤) = (1^st ‘𝑈))
2		clwlkclwwlkf.a	. . . . . 6 ⊢ 𝐴 = (1^st ‘𝑈)
3	1, 2	eqtr4di 2789	. . . . 5 ⊢ (𝑤 = 𝑈 → (1^st ‘𝑤) = 𝐴)
4	3	fveq2d 6895	. . . 4 ⊢ (𝑤 = 𝑈 → (♯‘(1^st ‘𝑤)) = (♯‘𝐴))
5	4	breq2d 5160	. . 3 ⊢ (𝑤 = 𝑈 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘𝐴)))
6		clwlkclwwlkf.c	. . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
7	5, 6	elrab2 3686	. 2 ⊢ (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8		clwlkwlk 29300	. . . 4 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9		wlkop 29153	. . . . 5 ⊢ (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
10		clwlkclwwlkf.b	. . . . . . . 8 ⊢ 𝐵 = (2^nd ‘𝑈)
11	2, 10	opeq12i 4878	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
12	11	eqeq2i 2744	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
13		eleq1 2820	. . . . . . 7 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺)))
14		df-br 5149	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺))
15		isclwlk 29298	. . . . . . . . 9 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))))
16		wlkcl 29140	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐴) ∈ ℕ₀)
17		elnnnn0c 12522	. . . . . . . . . . . . . . . 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 704	. . . . . . . . . . . . . 14 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ 1 ≤ (♯‘𝐴)))
20	19	bicomd 222	. . . . . . . . . . . . 13 ⊢ (𝐴(Walks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
22	21	pm5.32i 574	. . . . . . . . . . 11 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
23		df-3an 1088	. . . . . . . . . . 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 237	. . . . . . . . . 10 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
25	24	ex 412	. . . . . . . . 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 216	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
27	14, 26	sylbir 234	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
28	13, 27	syl6bi 253	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
29	12, 28	sylbir 234	. . . . 5 ⊢ (𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
30	9, 29	syl 17	. . . 4 ⊢ (𝑈 ∈ (Walks‘𝐺) → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
31	8, 30	mpcom 38	. . 3 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
32	31	imp 406	. 2 ⊢ ((𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
33	7, 32	sylbi 216	1 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {crab 3431 ⟨cop 4634 class class class wbr 5148 ‘cfv 6543 1^st c1st 7977 2^nd c2nd 7978 0cc0 11114 1c1 11115 ≤ cle 11254 ℕcn 12217 ℕ₀cn0 12477 ♯chash 14295 Walkscwlks 29121 ClWalkscclwlks 29295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-wlks 29124 df-clwlks 29296

This theorem is referenced by: clwlkclwwlkf1lem2 29526 clwlkclwwlkf1lem3 29527 clwlkclwwlkf 29529 clwlkclwwlkf1 29531

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