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Mirrors > Home > MPE Home > Th. List > crctcshlem2 | Structured version Visualization version GIF version |
Description: Lemma for crctcsh 26772. (Contributed by AV, 10-Mar-2021.) |
Ref | Expression |
---|---|
crctcsh.v | ⊢ 𝑉 = (Vtx‘𝐺) |
crctcsh.i | ⊢ 𝐼 = (iEdg‘𝐺) |
crctcsh.d | ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) |
crctcsh.n | ⊢ 𝑁 = (#‘𝐹) |
crctcsh.s | ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) |
crctcsh.h | ⊢ 𝐻 = (𝐹 cyclShift 𝑆) |
Ref | Expression |
---|---|
crctcshlem2 | ⊢ (𝜑 → (#‘𝐻) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crctcsh.d | . . . 4 ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) | |
2 | crctiswlk 26747 | . . . 4 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
3 | crctcsh.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 3 | wlkf 26566 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
5 | 1, 2, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) |
6 | crctcsh.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) | |
7 | elfzoelz 12509 | . . . 4 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℤ) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℤ) |
9 | cshwlen 13591 | . . 3 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑆 ∈ ℤ) → (#‘(𝐹 cyclShift 𝑆)) = (#‘𝐹)) | |
10 | 5, 8, 9 | syl2anc 694 | . 2 ⊢ (𝜑 → (#‘(𝐹 cyclShift 𝑆)) = (#‘𝐹)) |
11 | crctcsh.h | . . 3 ⊢ 𝐻 = (𝐹 cyclShift 𝑆) | |
12 | 11 | fveq2i 6232 | . 2 ⊢ (#‘𝐻) = (#‘(𝐹 cyclShift 𝑆)) |
13 | crctcsh.n | . 2 ⊢ 𝑁 = (#‘𝐹) | |
14 | 10, 12, 13 | 3eqtr4g 2710 | 1 ⊢ (𝜑 → (#‘𝐻) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ℤcz 11415 ..^cfzo 12504 #chash 13157 Word cword 13323 cyclShift ccsh 13580 Vtxcvtx 25919 iEdgciedg 25920 Walkscwlks 26548 Circuitsccrcts 26735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1033 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-hash 13158 df-word 13331 df-concat 13333 df-substr 13335 df-csh 13581 df-wlks 26551 df-trls 26645 df-crcts 26737 |
This theorem is referenced by: crctcshwlkn0 26769 crctcsh 26772 eucrctshift 27221 |
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