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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 29384 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(𝐴(PathsOn‘𝐺)𝐴)〈“𝐴𝐴”〉) |
3 | pthonispth 28983 | . . 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 28308 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺)) |
6 | s2fv1 14835 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘1) = 𝐴) | |
7 | s1len 14552 | . . . . . 6 ⊢ (♯‘〈“𝐽”〉) = 1 | |
8 | 7 | fveq2i 6891 | . . . . 5 ⊢ (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)) = (〈“𝐴𝐴”〉‘1) |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)) = (〈“𝐴𝐴”〉‘1)) |
10 | s2fv0 14834 | . . . 4 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐴”〉‘0) = 𝐴) | |
11 | 6, 9, 10 | 3eqtr4rd 2784 | . . 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 29028 | . 2 ⊢ (〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉 ↔ (〈“𝐽”〉(Paths‘𝐺)〈“𝐴𝐴”〉 ∧ (〈“𝐴𝐴”〉‘0) = (〈“𝐴𝐴”〉‘(♯‘〈“𝐽”〉)))) | |
14 | 4, 12, 13 | sylanbrc 584 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {csn 4627 class class class wbr 5147 dom cdm 5675 ‘cfv 6540 (class class class)co 7404 0cc0 11106 1c1 11107 ♯chash 14286 〈“cs1 14541 〈“cs2 14788 Vtxcvtx 28236 iEdgciedg 28237 UHGraphcuhgr 28296 Pathscpths 28949 PathsOncpthson 28951 Cyclesccycls 29022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-uhgr 28298 df-wlks 28836 df-wlkson 28837 df-trls 28929 df-trlson 28930 df-pths 28953 df-pthson 28955 df-cycls 29024 |
This theorem is referenced by: loop1cycl 34066 |
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