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Mirrors > Home > MPE Home > Th. List > clwlkclwwlkfolem | Structured version Visualization version GIF version |
Description: Lemma for clwlkclwwlkfo 27779. (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 1133 | . 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) | |
2 | wrdlenccats1lenm1 13968 | . . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1) = (♯‘𝑊)) | |
3 | 2 | eqcomd 2825 | . . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) = ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1)) |
4 | 3 | breq2d 5069 | . . . . 5 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (1 ≤ (♯‘𝑊) ↔ 1 ≤ ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1))) |
5 | 4 | biimpa 479 | . . . 4 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊)) → 1 ≤ ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1)) |
6 | 5 | 3adant3 1127 | . . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 1 ≤ ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1)) |
7 | df-br 5058 | . . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉) ↔ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) | |
8 | clwlkiswlk 27547 | . . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉) → 𝑓(Walks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉)) | |
9 | wlklenvm1 27395 | . . . . . 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 237 | . . . 4 ⊢ (〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺) → (♯‘𝑓) = ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1)) |
12 | 11 | 3ad2ant3 1130 | . . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1)) |
13 | 6, 12 | breqtrrd 5085 | . 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓)) |
14 | vex 3496 | . . . . . 6 ⊢ 𝑓 ∈ V | |
15 | ovex 7181 | . . . . . 6 ⊢ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ V | |
16 | 14, 15 | op1std 7691 | . . . . 5 ⊢ (𝑐 = 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 → (1st ‘𝑐) = 𝑓) |
17 | 16 | fveq2d 6667 | . . . 4 ⊢ (𝑐 = 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 → (♯‘(1st ‘𝑐)) = (♯‘𝑓)) |
18 | 17 | breq2d 5069 | . . 3 ⊢ (𝑐 = 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 → (1 ≤ (♯‘(1st ‘𝑐)) ↔ 1 ≤ (♯‘𝑓))) |
19 | clwlkclwwlkf.c | . . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} | |
20 | 2fveq3 6668 | . . . . . 6 ⊢ (𝑤 = 𝑐 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑐))) | |
21 | 20 | breq2d 5069 | . . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑐)))) |
22 | 21 | cbvrabv 3490 | . . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑐))} |
23 | 19, 22 | eqtri 2842 | . . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑐))} |
24 | 18, 23 | elrab2 3681 | . 2 ⊢ (〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ 𝐶 ↔ (〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝑓))) |
25 | 1, 13, 24 | sylanbrc 585 | 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 1082 = wceq 1531 ∈ wcel 2108 {crab 3140 〈cop 4565 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 1st c1st 7679 0cc0 10529 1c1 10530 ≤ cle 10668 − cmin 10862 ♯chash 13682 Word cword 13853 ++ cconcat 13914 〈“cs1 13941 Vtxcvtx 26773 Walkscwlks 27370 ClWalkscclwlks 27543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-wlks 27373 df-clwlks 27544 |
This theorem is referenced by: clwlkclwwlkfo 27779 |
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