Theorem clwlkclwwlkflem 28948

Description: Lemma for clwlkclwwlkf 28952. (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 6842	. . . . . 6 ⊢ (𝑤 = 𝑈 → (1st ‘𝑤) = (1st ‘𝑈))
2		clwlkclwwlkf.a	. . . . . 6 ⊢ 𝐴 = (1st ‘𝑈)
3	1, 2	eqtr4di 2794	. . . . 5 ⊢ (𝑤 = 𝑈 → (1st ‘𝑤) = 𝐴)
4	3	fveq2d 6846	. . . 4 ⊢ (𝑤 = 𝑈 → (♯‘(1st ‘𝑤)) = (♯‘𝐴))
5	4	breq2d 5117	. . 3 ⊢ (𝑤 = 𝑈 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘𝐴)))
6		clwlkclwwlkf.c	. . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
7	5, 6	elrab2 3648	. 2 ⊢ (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8		clwlkwlk 28723	. . . 4 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9		wlkop 28576	. . . . 5 ⊢ (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
10		clwlkclwwlkf.b	. . . . . . . 8 ⊢ 𝐵 = (2nd ‘𝑈)
11	2, 10	opeq12i 4835	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
12	11	eqeq2i 2749	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
13		eleq1 2825	. . . . . . 7 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺)))
14		df-br 5106	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺))
15		isclwlk 28721	. . . . . . . . 9 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))))
16		wlkcl 28563	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐴) ∈ ℕ0)
17		elnnnn0c 12458	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
22	21	pm5.32i 575	. . . . . . . . . . 11 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
23		df-3an 1089	. . . . . . . . . . 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 413	. . . . . . . . 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 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 = ⟨(1st ‘𝑈), (2nd ‘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 407	. 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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3407  ⟨cop 4592   class class class wbr 5105  ‘cfv 6496  1^st c1st 7919  2^nd c2nd 7920  0cc0 11051  1c1 11052   ≤ cle 11190  ℕcn 12153  ℕ₀cn0 12413  ♯chash 14230  Walkscwlks 28544  ClWalkscclwlks 28718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-wlks 28547  df-clwlks 28719

This theorem is referenced by:  clwlkclwwlkf1lem2  28949  clwlkclwwlkf1lem3  28950  clwlkclwwlkf  28952  clwlkclwwlkf1  28954