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Mirrors > Home > MPE Home > Th. List > Mathboxes > av-numclwwlk5lem | Structured version Visualization version GIF version |
Description: Lemma for av-numclwwlk5 41542. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.) |
Ref | Expression |
---|---|
av-numclwwlk4.v | ⊢ 𝑉 = (Vtx‘𝐺) |
av-numclwwlk4.f | ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
Ref | Expression |
---|---|
av-numclwwlk5lem | ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → ((#‘(𝑋𝐹2)) mod 2) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | av-numclwwlk4.f | . . . 4 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣}) | |
2 | av-numclwwlk4.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqid 2610 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 2, 3 | av-numclwwlkovf2num 41516 | . . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (#‘(𝑋𝐹2)) = 𝐾) |
5 | 4 | 3adant3 1074 | . 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (#‘(𝑋𝐹2)) = 𝐾) |
6 | oveq1 6556 | . . . . 5 ⊢ ((#‘(𝑋𝐹2)) = 𝐾 → ((#‘(𝑋𝐹2)) mod 2) = (𝐾 mod 2)) | |
7 | 6 | ad2antrr 758 | . . . 4 ⊢ ((((#‘(𝑋𝐹2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) ∧ 2 ∥ (𝐾 − 1)) → ((#‘(𝑋𝐹2)) mod 2) = (𝐾 mod 2)) |
8 | 2prm 15243 | . . . . . . . . 9 ⊢ 2 ∈ ℙ | |
9 | nn0z 11277 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℤ) | |
10 | modprm1div 15340 | . . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝐾 ∈ ℤ) → ((𝐾 mod 2) = 1 ↔ 2 ∥ (𝐾 − 1))) | |
11 | 8, 9, 10 | sylancr 694 | . . . . . . . 8 ⊢ (𝐾 ∈ ℕ0 → ((𝐾 mod 2) = 1 ↔ 2 ∥ (𝐾 − 1))) |
12 | 11 | biimprd 237 | . . . . . . 7 ⊢ (𝐾 ∈ ℕ0 → (2 ∥ (𝐾 − 1) → (𝐾 mod 2) = 1)) |
13 | 12 | 3ad2ant3 1077 | . . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → (𝐾 mod 2) = 1)) |
14 | 13 | adantl 481 | . . . . 5 ⊢ (((#‘(𝑋𝐹2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) → (2 ∥ (𝐾 − 1) → (𝐾 mod 2) = 1)) |
15 | 14 | imp 444 | . . . 4 ⊢ ((((#‘(𝑋𝐹2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) ∧ 2 ∥ (𝐾 − 1)) → (𝐾 mod 2) = 1) |
16 | 7, 15 | eqtrd 2644 | . . 3 ⊢ ((((#‘(𝑋𝐹2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) ∧ 2 ∥ (𝐾 − 1)) → ((#‘(𝑋𝐹2)) mod 2) = 1) |
17 | 16 | ex 449 | . 2 ⊢ (((#‘(𝑋𝐹2)) = 𝐾 ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0)) → (2 ∥ (𝐾 − 1) → ((#‘(𝑋𝐹2)) mod 2) = 1)) |
18 | 5, 17 | mpancom 700 | 1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → ((#‘(𝑋𝐹2)) mod 2) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 0cc0 9815 1c1 9816 − cmin 10145 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℤcz 11254 mod cmo 12530 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 Vtxcvtx 25673 Edgcedga 25792 RegUSGraph crusgr 40756 ClWWalkSN cclwwlksn 41184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-nbgr 40554 df-vtxdg 40682 df-rgr 40757 df-rusgr 40758 df-clwwlks 41185 df-clwwlksn 41186 |
This theorem is referenced by: av-numclwwlk5 41542 |
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