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Mirrors > Home > MPE Home > Th. List > clwlkl1loop | Structured version Visualization version GIF version |
Description: A closed walk of length 1 is a loop. (Contributed by AV, 22-Apr-2021.) |
Ref | Expression |
---|---|
clwlkl1loop | ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(ClWalks‘𝐺)𝑃 ∧ (♯‘𝐹) = 1) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isclwlk 27726 | . . 3 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
2 | fveq2 6686 | . . . . . . 7 ⊢ ((♯‘𝐹) = 1 → (𝑃‘(♯‘𝐹)) = (𝑃‘1)) | |
3 | 2 | eqeq2d 2750 | . . . . . 6 ⊢ ((♯‘𝐹) = 1 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘1))) |
4 | 3 | anbi2d 632 | . . . . 5 ⊢ ((♯‘𝐹) = 1 → ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)))) |
5 | simp2r 1201 | . . . . . . 7 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) ∧ Fun (iEdg‘𝐺)) → (𝑃‘0) = (𝑃‘1)) | |
6 | simp3 1139 | . . . . . . . 8 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) ∧ Fun (iEdg‘𝐺)) → Fun (iEdg‘𝐺)) | |
7 | simp2l 1200 | . . . . . . . 8 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) ∧ Fun (iEdg‘𝐺)) → 𝐹(Walks‘𝐺)𝑃) | |
8 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) → (𝑃‘0) = (𝑃‘1)) | |
9 | 8 | anim2i 620 | . . . . . . . . 9 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1))) → ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) |
10 | 9 | 3adant3 1133 | . . . . . . . 8 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) ∧ Fun (iEdg‘𝐺)) → ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) |
11 | wlkl1loop 27591 | . . . . . . . 8 ⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)) | |
12 | 6, 7, 10, 11 | syl21anc 837 | . . . . . . 7 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) ∧ Fun (iEdg‘𝐺)) → {(𝑃‘0)} ∈ (Edg‘𝐺)) |
13 | 5, 12 | jca 515 | . . . . . 6 ⊢ (((♯‘𝐹) = 1 ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) ∧ Fun (iEdg‘𝐺)) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))) |
14 | 13 | 3exp 1120 | . . . . 5 ⊢ ((♯‘𝐹) = 1 → ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘1)) → (Fun (iEdg‘𝐺) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))))) |
15 | 4, 14 | sylbid 243 | . . . 4 ⊢ ((♯‘𝐹) = 1 → ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (Fun (iEdg‘𝐺) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))))) |
16 | 15 | com13 88 | . . 3 ⊢ (Fun (iEdg‘𝐺) → ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((♯‘𝐹) = 1 → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))))) |
17 | 1, 16 | syl5bi 245 | . 2 ⊢ (Fun (iEdg‘𝐺) → (𝐹(ClWalks‘𝐺)𝑃 → ((♯‘𝐹) = 1 → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))))) |
18 | 17 | 3imp 1112 | 1 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(ClWalks‘𝐺)𝑃 ∧ (♯‘𝐹) = 1) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {csn 4526 class class class wbr 5040 Fun wfun 6343 ‘cfv 6349 0cc0 10627 1c1 10628 ♯chash 13794 iEdgciedg 26954 Edgcedg 27004 Walkscwlks 27550 ClWalkscclwlks 27723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-pm 8452 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-fzo 13137 df-hash 13795 df-word 13968 df-edg 27005 df-wlks 27553 df-clwlks 27724 |
This theorem is referenced by: (None) |
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