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Theorem clwlkclwwlkfolem 29525

Description: Lemma for clwlkclwwlkfo 29527. (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 14578	. . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1) = (♯‘𝑊))
3	2	eqcomd 2736	. . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
4	3	breq2d 5161	. . . . 5 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (1 ≤ (♯‘𝑊) ↔ 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1)))
5	4	biimpa 475	. . . 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 5150	. . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) ↔ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
8		clwlkiswlk 29296	. . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → 𝑓(Walks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩))
9		wlklenvm1 29144	. . . . . 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 5177	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓))
14		vex 3476	. . . . . 6 ⊢ 𝑓 ∈ V
15		ovex 7446	. . . . . 6 ⊢ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ V
16	14, 15	op1std 7989	. . . . 5 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (1^st ‘𝑐) = 𝑓)
17	16	fveq2d 6896	. . . 4 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (♯‘(1^st ‘𝑐)) = (♯‘𝑓))
18	17	breq2d 5161	. . 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 6897	. . . . . 6 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
21	20	breq2d 5161	. . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
22	21	cbvrabv 3440	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
23	19, 22	eqtri 2758	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
24	18, 23	elrab2 3687	. 2 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝑓)))
25	1, 13, 24	sylanbrc 581	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 2104 {crab 3430 ⟨cop 4635 class class class wbr 5149 ‘cfv 6544 (class class class)co 7413 1^st c1st 7977 0cc0 11114 1c1 11115 ≤ cle 11255 − cmin 11450 ♯chash 14296 Word cword 14470 ++ cconcat 14526 ⟨“cs1 14551 Vtxcvtx 28521 Walkscwlks 29118 ClWalkscclwlks 29292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 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 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 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 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-concat 14527 df-s1 14552 df-wlks 29121 df-clwlks 29293

This theorem is referenced by: clwlkclwwlkfo 29527

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