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Mirrors > Home > MPE Home > Th. List > numclwwlk7lem | Structured version Visualization version GIF version |
Description: Lemma for numclwwlk7 27862, frgrreggt1 27864 and frgrreg 27865: If a finite, nonempty friendship graph is 𝐾-regular, the 𝐾 is a nonnegative integer. (Contributed by AV, 3-Jun-2021.) |
Ref | Expression |
---|---|
numclwwlk7lem.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
numclwwlk7lem | ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numclwwlk7lem.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | finrusgrfusgr 27030 | . . 3 ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
3 | 2 | ad2ant2rl 745 | . 2 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐺 ∈ FinUSGraph) |
4 | simpll 763 | . 2 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐺RegUSGraph𝐾) | |
5 | simprl 767 | . 2 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝑉 ≠ ∅) | |
6 | 1 | frusgrnn0 27036 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0) |
7 | 3, 4, 5, 6 | syl3anc 1364 | 1 ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∅c0 4211 class class class wbr 4962 ‘cfv 6225 Fincfn 8357 ℕ0cn0 11745 Vtxcvtx 26464 FinUSGraphcfusgr 26781 RegUSGraphcrusgr 27021 FriendGraph cfrgr 27727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-dju 9176 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-xnn0 11816 df-z 11830 df-uz 12094 df-xadd 12358 df-fz 12743 df-hash 13541 df-vtx 26466 df-iedg 26467 df-edg 26516 df-uhgr 26526 df-upgr 26550 df-umgr 26551 df-uspgr 26618 df-usgr 26619 df-fusgr 26782 df-vtxdg 26931 df-rgr 27022 df-rusgr 27023 |
This theorem is referenced by: numclwwlk7 27862 frgrreggt1 27864 frgrreg 27865 |
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