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Theorem clwlkclwwlkfolem 30094

Description: Lemma for clwlkclwwlkfo 30096. (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 1139	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
2		wrdlenccats1lenm1 14558	. . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1) = (♯‘𝑊))
3	2	eqcomd 2743	. . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
4	3	breq2d 5112	. . . . 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 1133	. . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
7		df-br 5101	. . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) ↔ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
8		clwlkiswlk 29859	. . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → 𝑓(Walks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩))
9		wlklenvm1 29707	. . . . . 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 235	. . . 4 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
12	11	3ad2ant3 1136	. . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
13	6, 12	breqtrrd 5128	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓))
14		vex 3446	. . . . . 6 ⊢ 𝑓 ∈ V
15		ovex 7401	. . . . . 6 ⊢ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ V
16	14, 15	op1std 7953	. . . . 5 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (1^st ‘𝑐) = 𝑓)
17	16	fveq2d 6846	. . . 4 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (♯‘(1^st ‘𝑐)) = (♯‘𝑓))
18	17	breq2d 5112	. . 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 6847	. . . . . 6 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
21	20	breq2d 5112	. . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
22	21	cbvrabv 3411	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
23	19, 22	eqtri 2760	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
24	18, 23	elrab2 3651	. 2 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝑓)))
25	1, 13, 24	sylanbrc 584	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 1087 = wceq 1542 ∈ wcel 2114 {crab 3401 ⟨cop 4588 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 0cc0 11038 1c1 11039 ≤ cle 11179 − cmin 11376 ♯chash 14265 Word cword 14448 ++ cconcat 14505 ⟨“cs1 14531 Vtxcvtx 29081 Walkscwlks 29682 ClWalkscclwlks 29855

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-concat 14506 df-s1 14532 df-wlks 29685 df-clwlks 29856

This theorem is referenced by: clwlkclwwlkfo 30096

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