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Mirrors > Home > MPE Home > Th. List > eclclwwlkn1 | Structured version Visualization version GIF version |
Description: An equivalence class according to ∼. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlkn.w | ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) |
erclwwlkn.r | ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} |
Ref | Expression |
---|---|
eclclwwlkn1 | ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsecl 8779 | . 2 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦})) | |
2 | erclwwlkn.w | . . . . . . . . 9 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) | |
3 | erclwwlkn.r | . . . . . . . . 9 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
4 | 2, 3 | erclwwlknsym 29854 | . . . . . . . 8 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥) |
5 | 2, 3 | erclwwlknsym 29854 | . . . . . . . 8 ⊢ (𝑦 ∼ 𝑥 → 𝑥 ∼ 𝑦) |
6 | 4, 5 | impbii 208 | . . . . . . 7 ⊢ (𝑥 ∼ 𝑦 ↔ 𝑦 ∼ 𝑥) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∼ 𝑦 ↔ 𝑦 ∼ 𝑥)) |
8 | 7 | abbidv 2796 | . . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ 𝑥 ∼ 𝑦} = {𝑦 ∣ 𝑦 ∼ 𝑥}) |
9 | 2, 3 | erclwwlkneq 29851 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
10 | 9 | el2v 3477 | . . . . . . 7 ⊢ (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))) |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
12 | 11 | abbidv 2796 | . . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ 𝑦 ∼ 𝑥} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))}) |
13 | 3anan12 1094 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) | |
14 | ibar 528 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))) | |
15 | 14 | bicomd 222 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → ((𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))) ↔ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
16 | 15 | adantl 481 | . . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → ((𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))) ↔ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
17 | 13, 16 | bitrid 283 | . . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) ↔ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))) |
18 | 17 | abbidv 2796 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))}) |
19 | df-rab 3428 | . . . . . 6 ⊢ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))} | |
20 | 18, 19 | eqtr4di 2785 | . . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))} = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
21 | 8, 12, 20 | 3eqtrd 2771 | . . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ 𝑥 ∼ 𝑦} = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}) |
22 | 21 | eqeq2d 2738 | . . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → (𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦} ↔ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
23 | 22 | rexbidva 3171 | . 2 ⊢ (𝐵 ∈ 𝑋 → (∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦} ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
24 | 1, 23 | bitrd 279 | 1 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {cab 2704 ∃wrex 3065 {crab 3427 Vcvv 3469 class class class wbr 5142 {copab 5204 (class class class)co 7414 / cqs 8715 0cc0 11124 ...cfz 13502 cyclShift ccsh 14756 ClWWalksN cclwwlkn 29808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-ec 8718 df-qs 8722 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-hash 14308 df-word 14483 df-concat 14539 df-substr 14609 df-pfx 14639 df-csh 14757 df-clwwlk 29766 df-clwwlkn 29809 |
This theorem is referenced by: eleclclwwlkn 29860 hashecclwwlkn1 29861 umgrhashecclwwlk 29862 |
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