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Mirrors > Home > MPE Home > Th. List > clwlkclwwlkfolem | Structured version Visualization version GIF version |
Description: Lemma for clwlkclwwlkfo 27785. (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 13971 | . . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1) = (♯‘𝑊)) | |
3 | 2 | eqcomd 2826 | . . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) = ((♯‘(𝑊 ++ 〈“(𝑊‘0)”〉)) − 1)) |
4 | 3 | breq2d 5071 | . . . . 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 5060 | . . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉) ↔ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) | |
8 | clwlkiswlk 27553 | . . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉) → 𝑓(Walks‘𝐺)(𝑊 ++ 〈“(𝑊‘0)”〉)) | |
9 | wlklenvm1 27401 | . . . . . 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 5087 | . 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓)) |
14 | vex 3494 | . . . . . 6 ⊢ 𝑓 ∈ V | |
15 | ovex 7182 | . . . . . 6 ⊢ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ V | |
16 | 14, 15 | op1std 7692 | . . . . 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 5071 | . . 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 5071 | . . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘(1st ‘𝑐)))) |
22 | 21 | cbvrabv 3488 | . . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑐))} |
23 | 19, 22 | eqtri 2843 | . . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑐))} |
24 | 18, 23 | elrab2 3679 | . 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 1536 ∈ wcel 2113 {crab 3141 〈cop 4566 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 1st c1st 7680 0cc0 10530 1c1 10531 ≤ cle 10669 − cmin 10863 ♯chash 13687 Word cword 13858 ++ cconcat 13917 〈“cs1 13944 Vtxcvtx 26779 Walkscwlks 27376 ClWalkscclwlks 27549 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
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 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-concat 13918 df-s1 13945 df-wlks 27379 df-clwlks 27550 |
This theorem is referenced by: clwlkclwwlkfo 27785 |
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