Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 28188 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(𝐴(PathsOn‘𝐺)𝐴)〈“𝐴𝐴”〉) |
3 | pthonispth 27787 | . . 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 27113 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺)) |
6 | s2fv1 14418 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘1) = 𝐴) | |
7 | s1len 14128 | . . . . . 6 ⊢ (♯‘〈“𝐽”〉) = 1 | |
8 | 7 | fveq2i 6698 | . . . . 5 ⊢ (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)) = (〈“𝐴𝐴”〉‘1) |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)) = (〈“𝐴𝐴”〉‘1)) |
10 | s2fv0 14417 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘0) = 𝐴) | |
11 | 6, 9, 10 | 3eqtr4rd 2782 | . . 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 27832 | . 2 ⊢ (〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉 ↔ (〈“𝐽”〉(Paths‘𝐺)〈“𝐴𝐴”〉 ∧ (〈“𝐴𝐴”〉‘0) = (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)))) | |
14 | 4, 12, 13 | sylanbrc 586 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 {csn 4527 class class class wbr 5039 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 ♯chash 13861 〈“cs1 14117 〈“cs2 14371 Vtxcvtx 27041 iEdgciedg 27042 UHGraphcuhgr 27101 Pathscpths 27753 PathsOncpthson 27755 Cyclesccycls 27826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-uhgr 27103 df-wlks 27641 df-wlkson 27642 df-trls 27734 df-trlson 27735 df-pths 27757 df-pthson 27759 df-cycls 27828 |
This theorem is referenced by: loop1cycl 32766 |
Copyright terms: Public domain | W3C validator |