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Mirrors > Home > MPE Home > Th. List > clwlkclwwlkfolem | Structured version Visualization version GIF version |
Description: Lemma for clwlkclwwlkfo 27938. (Contributed by AV, 25-May-2022.) |
Ref | Expression |
---|---|
clwlkclwwlkf.c | ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} |
Ref | Expression |
---|---|
clwlkclwwlkfolem | ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ 𝐶) |
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 14058 | . . . . . . 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 5039 | . . . . 5 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (1 ≤ (♯‘𝑊) ↔ 1 ≤ ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1))) |
5 | 4 | biimpa 480 | . . . 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 5028 | . . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉) ↔ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) | |
8 | clwlkiswlk 27707 | . . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉) → 𝑓(Walks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉)) | |
9 | wlklenvm1 27555 | . . . . . 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 238 | . . . 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 5055 | . 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓)) |
14 | vex 3401 | . . . . . 6 ⊢ 𝑓 ∈ V | |
15 | ovex 7197 | . . . . . 6 ⊢ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ V | |
16 | 14, 15 | op1std 7717 | . . . . 5 ⊢ (𝑐 = 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 → (1st ‘𝑐) = 𝑓) |
17 | 16 | fveq2d 6672 | . . . 4 ⊢ (𝑐 = 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 → (♯‘(1st ‘𝑐)) = (♯‘𝑓)) |
18 | 17 | breq2d 5039 | . . 3 ⊢ (𝑐 = 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 → (1 ≤ (♯‘(1st ‘𝑐)) ↔ 1 ≤ (♯‘𝑓))) |
19 | clwlkclwwlkf.c | . . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} | |
20 | 2fveq3 6673 | . . . . . 6 ⊢ (𝑤 = 𝑐 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑐))) | |
21 | 20 | breq2d 5039 | . . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑐)))) |
22 | 21 | cbvrabv 3392 | . . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑐))} |
23 | 19, 22 | eqtri 2761 | . . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑐))} |
24 | 18, 23 | elrab2 3588 | . 2 ⊢ (〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ 𝐶 ↔ (〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝑓))) |
25 | 1, 13, 24 | sylanbrc 586 | 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 1088 = wceq 1542 ∈ wcel 2113 {crab 3057 〈cop 4519 class class class wbr 5027 ‘cfv 6333 (class class class)co 7164 1st c1st 7705 0cc0 10608 1c1 10609 ≤ cle 10747 − cmin 10941 ♯chash 13775 Word cword 13948 ++ cconcat 14004 〈“cs1 14031 Vtxcvtx 26933 Walkscwlks 27530 ClWalkscclwlks 27703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-fzo 13118 df-hash 13776 df-word 13949 df-concat 14005 df-s1 14032 df-wlks 27533 df-clwlks 27704 |
This theorem is referenced by: clwlkclwwlkfo 27938 |
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