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Theorem clwlkclwwlkfolem 28272

Description: Lemma for clwlkclwwlkfo 28274. (Contributed by AV, 25-May-2022.)

**Hypothesis**
Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}

**Assertion**
Ref	Expression
clwlkclwwlkfolem	⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶)

Distinct variable group: 𝑤,𝐺,𝑓 Allowed substitution hints: 𝐶(𝑤,𝑓) 𝑊(𝑤,𝑓)

Proof of Theorem clwlkclwwlkfolem Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1136	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
2		wrdlenccats1lenm1 14255	. . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1) = (♯‘𝑊))
3	2	eqcomd 2744	. . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
4	3	breq2d 5082	. . . . 5 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (1 ≤ (♯‘𝑊) ↔ 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1)))
5	4	biimpa 476	. . . 4 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊)) → 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
6	5	3adant3 1130	. . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
7		df-br 5071	. . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) ↔ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
8		clwlkiswlk 28043	. . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → 𝑓(Walks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩))
9		wlklenvm1 27891	. . . . . 6 ⊢ (𝑓(Walks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
10	8, 9	syl 17	. . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
11	7, 10	sylbir 234	. . . 4 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
12	11	3ad2ant3 1133	. . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
13	6, 12	breqtrrd 5098	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓))
14		vex 3426	. . . . . 6 ⊢ 𝑓 ∈ V
15		ovex 7288	. . . . . 6 ⊢ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ V
16	14, 15	op1std 7814	. . . . 5 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (1^st ‘𝑐) = 𝑓)
17	16	fveq2d 6760	. . . 4 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (♯‘(1^st ‘𝑐)) = (♯‘𝑓))
18	17	breq2d 5082	. . 3 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ (♯‘𝑓)))
19		clwlkclwwlkf.c	. . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
20		2fveq3 6761	. . . . . 6 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
21	20	breq2d 5082	. . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
22	21	cbvrabv 3416	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
23	19, 22	eqtri 2766	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
24	18, 23	elrab2 3620	. 2 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝑓)))
25	1, 13, 24	sylanbrc 582	1 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 {crab 3067 ⟨cop 4564 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 0cc0 10802 1c1 10803 ≤ cle 10941 − cmin 11135 ♯chash 13972 Word cword 14145 ++ cconcat 14201 ⟨“cs1 14228 Vtxcvtx 27269 Walkscwlks 27866 ClWalkscclwlks 28039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-wlks 27869 df-clwlks 28040

This theorem is referenced by: clwlkclwwlkfo 28274

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