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Mirrors > Home > MPE Home > Th. List > frgrregord13 | Structured version Visualization version GIF version |
Description: If a nonempty finite friendship graph is ðŸ-regular, then it must have order 1 or 3. Special case of frgrregord013 30179. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.) |
Ref | Expression |
---|---|
frgrreggt1.v | ⢠ð = (Vtxâðº) |
Ref | Expression |
---|---|
frgrregord13 | ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 1 âš (â¯âð) = 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1189 | . . 3 ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ðº â FriendGraph ) | |
2 | simpl2 1190 | . . 3 ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ð â Fin) | |
3 | simpr 484 | . . 3 ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ðº RegUSGraph ðŸ) | |
4 | frgrreggt1.v | . . . 4 ⢠ð = (Vtxâðº) | |
5 | 4 | frgrregord013 30179 | . . 3 ⢠((ðº â FriendGraph â§ ð â Fin â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 0 âš (â¯âð) = 1 âš (â¯âð) = 3)) |
6 | 1, 2, 3, 5 | syl3anc 1369 | . 2 ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 0 âš (â¯âð) = 1 âš (â¯âð) = 3)) |
7 | hasheq0 14340 | . . . . . . . . 9 ⢠(ð â Fin â ((â¯âð) = 0 â ð = â )) | |
8 | eqneqall 2946 | . . . . . . . . 9 ⢠(ð = â â (ð â â â ((â¯âð) = 1 âš (â¯âð) = 3))) | |
9 | 7, 8 | syl6bi 253 | . . . . . . . 8 ⢠(ð â Fin â ((â¯âð) = 0 â (ð â â â ((â¯âð) = 1 âš (â¯âð) = 3)))) |
10 | 9 | com23 86 | . . . . . . 7 ⢠(ð â Fin â (ð â â â ((â¯âð) = 0 â ((â¯âð) = 1 âš (â¯âð) = 3)))) |
11 | 10 | a1i 11 | . . . . . 6 ⢠(ðº â FriendGraph â (ð â Fin â (ð â â â ((â¯âð) = 0 â ((â¯âð) = 1 âš (â¯âð) = 3))))) |
12 | 11 | 3imp 1109 | . . . . 5 ⢠((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â ((â¯âð) = 0 â ((â¯âð) = 1 âš (â¯âð) = 3))) |
13 | 12 | adantr 480 | . . . 4 ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 0 â ((â¯âð) = 1 âš (â¯âð) = 3))) |
14 | 13 | com12 32 | . . 3 ⢠((â¯âð) = 0 â (((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 1 âš (â¯âð) = 3))) |
15 | orc 866 | . . . 4 ⢠((â¯âð) = 1 â ((â¯âð) = 1 âš (â¯âð) = 3)) | |
16 | 15 | a1d 25 | . . 3 ⢠((â¯âð) = 1 â (((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 1 âš (â¯âð) = 3))) |
17 | olc 867 | . . . 4 ⢠((â¯âð) = 3 â ((â¯âð) = 1 âš (â¯âð) = 3)) | |
18 | 17 | a1d 25 | . . 3 ⢠((â¯âð) = 3 â (((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 1 âš (â¯âð) = 3))) |
19 | 14, 16, 18 | 3jaoi 1425 | . 2 ⢠(((â¯âð) = 0 âš (â¯âð) = 1 âš (â¯âð) = 3) â (((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 1 âš (â¯âð) = 3))) |
20 | 6, 19 | mpcom 38 | 1 ⢠(((ðº â FriendGraph â§ ð â Fin â§ ð â â ) â§ ðº RegUSGraph ðŸ) â ((â¯âð) = 1 âš (â¯âð) = 3)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 395 âš wo 846 âš w3o 1084 â§ w3a 1085 = wceq 1534 â wcel 2099 â wne 2935 â c0 4318 class class class wbr 5142 âcfv 6542 Fincfn 8953 0cc0 11124 1c1 11125 3c3 12284 â¯chash 14307 Vtxcvtx 28783 RegUSGraph crusgr 29344 FriendGraph cfrgr 30042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-ac2 10472 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-ec 8718 df-qs 8722 df-map 8836 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-oi 9519 df-dju 9910 df-card 9948 df-ac 10125 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-xnn0 12561 df-z 12575 df-uz 12839 df-rp 12993 df-xadd 13111 df-ico 13348 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-hash 14308 df-word 14483 df-lsw 14531 df-concat 14539 df-s1 14564 df-substr 14609 df-pfx 14639 df-reps 14737 df-csh 14757 df-s2 14817 df-s3 14818 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-sum 15651 df-dvds 16217 df-gcd 16455 df-prm 16628 df-phi 16720 df-vtx 28785 df-iedg 28786 df-edg 28835 df-uhgr 28845 df-ushgr 28846 df-upgr 28869 df-umgr 28870 df-uspgr 28937 df-usgr 28938 df-fusgr 29104 df-nbgr 29120 df-vtxdg 29254 df-rgr 29345 df-rusgr 29346 df-wlks 29387 df-wlkson 29388 df-trls 29480 df-trlson 29481 df-pths 29504 df-spths 29505 df-pthson 29506 df-spthson 29507 df-wwlks 29615 df-wwlksn 29616 df-wwlksnon 29617 df-wspthsn 29618 df-wspthsnon 29619 df-clwwlk 29766 df-clwwlkn 29809 df-clwwlknon 29872 df-conngr 29971 df-frgr 30043 |
This theorem is referenced by: frgrogt3nreg 30181 |
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