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Mirrors > Home > MPE Home > Th. List > crctcshlem3 | Structured version Visualization version GIF version |
Description: Lemma for crctcsh 27304. (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) |
crctcsh.q | ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) |
Ref | Expression |
---|---|
crctcshlem3 | ⊢ (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crctcsh.d | . . 3 ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) | |
2 | crctistrl 27278 | . . 3 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
3 | trliswlk 27179 | . . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
4 | wlkv 27091 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
5 | simp1 1116 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → 𝐺 ∈ V) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐺 ∈ V) |
7 | 1, 2, 6 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐺 ∈ V) |
8 | crctcsh.h | . . . 4 ⊢ 𝐻 = (𝐹 cyclShift 𝑆) | |
9 | 8 | ovexi 7003 | . . 3 ⊢ 𝐻 ∈ V |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
11 | crctcsh.q | . . . 4 ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) | |
12 | ovex 7002 | . . . . 5 ⊢ (0...𝑁) ∈ V | |
13 | 12 | mptex 6806 | . . . 4 ⊢ (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) ∈ V |
14 | 11, 13 | eqeltri 2856 | . . 3 ⊢ 𝑄 ∈ V |
15 | 14 | a1i 11 | . 2 ⊢ (𝜑 → 𝑄 ∈ V) |
16 | 7, 10, 15 | 3jca 1108 | 1 ⊢ (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 Vcvv 3409 ifcif 4344 class class class wbr 4923 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 0cc0 10329 + caddc 10332 ≤ cle 10469 − cmin 10664 ...cfz 12702 ..^cfzo 12843 ♯chash 13499 cyclShift ccsh 14001 Vtxcvtx 26478 iEdgciedg 26479 Walkscwlks 27075 Trailsctrls 27172 Circuitsccrcts 27267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ifp 1044 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-fzo 12844 df-hash 13500 df-word 13667 df-wlks 27078 df-trls 27174 df-crcts 27269 |
This theorem is referenced by: crctcshwlkn0 27301 |
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