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Mirrors > Home > MPE Home > Th. List > numclwwlk5lem | Structured version Visualization version GIF version |
Description: Lemma for numclwwlk5 28257. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.) (Revised by AV, 7-Mar-2022.) |
Ref | Expression |
---|---|
numclwwlk3.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
numclwwlk5lem | ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numclwwlk3.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | eleq2i 2842 | . . . 4 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺)) |
3 | clwwlknon2num 27974 | . . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾) | |
4 | 2, 3 | sylan2b 597 | . . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾) |
5 | 4 | 3adant3 1130 | . 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾) |
6 | oveq1 7150 | . . . . 5 ⊢ ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾 → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = (𝐾 mod 2)) | |
7 | 6 | ad2antrr 726 | . . . 4 ⊢ ((((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) ∧ 2 ∥ (𝐾 − 1)) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = (𝐾 mod 2)) |
8 | 2prm 16073 | . . . . . . . . 9 ⊢ 2 ∈ ℙ | |
9 | nn0z 12029 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℤ) | |
10 | modprm1div 16174 | . . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝐾 ∈ ℤ) → ((𝐾 mod 2) = 1 ↔ 2 ∥ (𝐾 − 1))) | |
11 | 8, 9, 10 | sylancr 591 | . . . . . . . 8 ⊢ (𝐾 ∈ ℕ0 → ((𝐾 mod 2) = 1 ↔ 2 ∥ (𝐾 − 1))) |
12 | 11 | biimprd 251 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → (2 ∥ (𝐾 − 1) → (𝐾 mod 2) = 1)) |
13 | 12 | 3ad2ant3 1133 | . . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → (𝐾 mod 2) = 1)) |
14 | 13 | adantl 486 | . . . . 5 ⊢ (((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) → (2 ∥ (𝐾 − 1) → (𝐾 mod 2) = 1)) |
15 | 14 | imp 411 | . . . 4 ⊢ ((((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) ∧ 2 ∥ (𝐾 − 1)) → (𝐾 mod 2) = 1) |
16 | 7, 15 | eqtrd 2794 | . . 3 ⊢ ((((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) ∧ 2 ∥ (𝐾 − 1)) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1) |
17 | 16 | ex 417 | . 2 ⊢ (((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) → (2 ∥ (𝐾 − 1) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1)) |
18 | 5, 17 | mpancom 688 | 1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 class class class wbr 5025 ‘cfv 6328 (class class class)co 7143 1c1 10561 − cmin 10893 2c2 11714 ℕ0cn0 11919 ℤcz 12005 mod cmo 13271 ♯chash 13725 ∥ cdvds 15640 ℙcprime 16052 Vtxcvtx 26873 RegUSGraph crusgr 27430 ClWWalksNOncclwwlknon 27956 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-sup 8924 df-inf 8925 df-dju 9348 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-n0 11920 df-xnn0 11992 df-z 12006 df-uz 12268 df-rp 12416 df-xadd 12534 df-fz 12925 df-fzo 13068 df-fl 13196 df-mod 13272 df-seq 13404 df-exp 13465 df-hash 13726 df-word 13899 df-lsw 13947 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-dvds 15641 df-prm 16053 df-edg 26925 df-uhgr 26935 df-ushgr 26936 df-upgr 26959 df-umgr 26960 df-uspgr 27027 df-usgr 27028 df-nbgr 27207 df-vtxdg 27340 df-rgr 27431 df-rusgr 27432 df-clwwlk 27851 df-clwwlkn 27894 df-clwwlknon 27957 |
This theorem is referenced by: numclwwlk5 28257 |
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