clwlkclwwlkflem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > clwlkclwwlkflem		Structured version Visualization version GIF version

Theorem clwlkclwwlkflem 29832

Description: Lemma for clwlkclwwlkf 29836. (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 6900	. . . . . 6 ⊢ (𝑤 = 𝑈 → (1^st ‘𝑤) = (1^st ‘𝑈))
2		clwlkclwwlkf.a	. . . . . 6 ⊢ 𝐴 = (1^st ‘𝑈)
3	1, 2	eqtr4di 2785	. . . . 5 ⊢ (𝑤 = 𝑈 → (1^st ‘𝑤) = 𝐴)
4	3	fveq2d 6904	. . . 4 ⊢ (𝑤 = 𝑈 → (♯‘(1^st ‘𝑤)) = (♯‘𝐴))
5	4	breq2d 5162	. . 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 3685	. 2 ⊢ (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8		clwlkwlk 29607	. . . 4 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9		wlkop 29460	. . . . 5 ⊢ (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
10		clwlkclwwlkf.b	. . . . . . . 8 ⊢ 𝐵 = (2^nd ‘𝑈)
11	2, 10	opeq12i 4881	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
12	11	eqeq2i 2740	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
13		eleq1 2816	. . . . . . 7 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺)))
14		df-br 5151	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺))
15		isclwlk 29605	. . . . . . . . 9 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))))
16		wlkcl 29447	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐴) ∈ ℕ₀)
17		elnnnn0c 12553	. . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . 12 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
22	21	pm5.32i 573	. . . . . . . . . . 11 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
23		df-3an 1086	. . . . . . . . . . 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 411	. . . . . . . . 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	biimtrdi 252	. . . . . 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 405	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3428 ⟨cop 4636 class class class wbr 5150 ‘cfv 6551 1^st c1st 7995 2^nd c2nd 7996 0cc0 11144 1c1 11145 ≤ cle 11285 ℕcn 12248 ℕ₀cn0 12508 ♯chash 14327 Walkscwlks 29428 ClWalkscclwlks 29602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-fzo 13666 df-hash 14328 df-word 14503 df-wlks 29431 df-clwlks 29603

This theorem is referenced by: clwlkclwwlkf1lem2 29833 clwlkclwwlkf1lem3 29834 clwlkclwwlkf 29836 clwlkclwwlkf1 29838

W3C validator

	
		OSZAR »