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Mirrors > Home > MPE Home > Th. List > crctcshlem3 | Structured version Visualization version GIF version |
Description: Lemma for crctcsh 29853. (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 29827 | . . 3 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
3 | trliswlk 29729 | . . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
4 | wlkv 29644 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
5 | simp1 1135 | . . . 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 7464 | . . 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 7463 | . . . . 5 ⊢ (0...𝑁) ∈ V | |
13 | 12 | mptex 7242 | . . . 4 ⊢ (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) ∈ V |
14 | 11, 13 | eqeltri 2834 | . . 3 ⊢ 𝑄 ∈ V |
15 | 14 | a1i 11 | . 2 ⊢ (𝜑 → 𝑄 ∈ V) |
16 | 7, 10, 15 | 3jca 1127 | 1 ⊢ (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ifcif 4530 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 0cc0 11152 + caddc 11155 ≤ cle 11293 − cmin 11489 ...cfz 13543 ..^cfzo 13690 ♯chash 14365 cyclShift ccsh 14822 Vtxcvtx 29027 iEdgciedg 29028 Walkscwlks 29628 Trailsctrls 29722 Circuitsccrcts 29816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-wlks 29631 df-trls 29724 df-crcts 29818 |
This theorem is referenced by: crctcshwlkn0 29850 |
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