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Mirrors > Home > MPE Home > Th. List > lp1cycl | Structured version Visualization version GIF version |
Description: A loop (which is an edge at index 𝐽) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
lppthon.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
lp1cycl | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lppthon.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | 1 | lppthon 27924 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(𝐴(PathsOn‘𝐺)𝐴)〈“𝐴𝐴”〉) |
3 | pthonispth 27521 | . . 3 ⊢ (〈“𝐽”〉(𝐴(PathsOn‘𝐺)𝐴)〈“𝐴𝐴”〉 → 〈“𝐽”〉(Paths‘𝐺)〈“𝐴𝐴”〉) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(Paths‘𝐺)〈“𝐴𝐴”〉) |
5 | 1 | lpvtx 26847 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺)) |
6 | s2fv1 14244 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘1) = 𝐴) | |
7 | s1len 13954 | . . . . . 6 ⊢ (♯‘〈“𝐽”〉) = 1 | |
8 | 7 | fveq2i 6668 | . . . . 5 ⊢ (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)) = (〈“𝐴𝐴”〉‘1) |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)) = (〈“𝐴𝐴”〉‘1)) |
10 | s2fv0 14243 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘0) = 𝐴) | |
11 | 6, 9, 10 | 3eqtr4rd 2867 | . . 3 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘0) = (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉))) |
12 | 5, 11 | syl 17 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → (〈“𝐴𝐴”〉‘0) = (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉))) |
13 | iscycl 27566 | . 2 ⊢ (〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉 ↔ (〈“𝐽”〉(Paths‘𝐺)〈“𝐴𝐴”〉 ∧ (〈“𝐴𝐴”〉‘0) = (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)))) | |
14 | 4, 12, 13 | sylanbrc 585 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {csn 4561 class class class wbr 5059 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 0cc0 10531 1c1 10532 ♯chash 13684 〈“cs1 13943 〈“cs2 14197 Vtxcvtx 26775 iEdgciedg 26776 UHGraphcuhgr 26835 Pathscpths 27487 PathsOncpthson 27489 Cyclesccycls 27560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-uhgr 26837 df-wlks 27375 df-wlkson 27376 df-trls 27468 df-trlson 27469 df-pths 27491 df-pthson 27493 df-cycls 27562 |
This theorem is referenced by: loop1cycl 32379 |
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