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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpm3cl2 | Structured version Visualization version GIF version |
Description: Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
cycpm3.c | ⊢ 𝐶 = (toCyc‘𝐷) |
cycpm3.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
cycpm3.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
cycpm3.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
cycpm3.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
cycpm3.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
cycpm3.1 | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
cycpm3.2 | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
cycpm3.3 | ⊢ (𝜑 → 𝐾 ≠ 𝐼) |
Ref | Expression |
---|---|
cycpm3cl2 | ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (𝐶 “ (◡♯ “ {3}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpm3.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
2 | cycpm3.c | . . . . 5 ⊢ 𝐶 = (toCyc‘𝐷) | |
3 | cycpm3.s | . . . . 5 ⊢ 𝑆 = (SymGrp‘𝐷) | |
4 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
5 | 2, 3, 4 | tocycf 31966 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
7 | 6 | ffnd 6669 | . 2 ⊢ (𝜑 → 𝐶 Fn {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
8 | id 22 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → 𝑤 = 〈“𝐼𝐽𝐾”〉) | |
9 | dmeq 5859 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → dom 𝑤 = dom 〈“𝐼𝐽𝐾”〉) | |
10 | eqidd 2737 | . . . 4 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → 𝐷 = 𝐷) | |
11 | 8, 9, 10 | f1eq123d 6776 | . . 3 ⊢ (𝑤 = 〈“𝐼𝐽𝐾”〉 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷)) |
12 | cycpm3.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
13 | cycpm3.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
14 | cycpm3.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
15 | 12, 13, 14 | s3cld 14761 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ Word 𝐷) |
16 | cycpm3.1 | . . . 4 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
17 | cycpm3.2 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
18 | cycpm3.3 | . . . 4 ⊢ (𝜑 → 𝐾 ≠ 𝐼) | |
19 | 12, 13, 14, 16, 17, 18 | s3f1 31803 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉:dom 〈“𝐼𝐽𝐾”〉–1-1→𝐷) |
20 | 11, 15, 19 | elrabd 3647 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
21 | s3clhash 31804 | . . 3 ⊢ 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3}) | |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽𝐾”〉 ∈ (◡♯ “ {3})) |
23 | 7, 20, 22 | fnfvimad 7184 | 1 ⊢ (𝜑 → (𝐶‘〈“𝐼𝐽𝐾”〉) ∈ (𝐶 “ (◡♯ “ {3}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {crab 3407 {csn 4586 ◡ccnv 5632 dom cdm 5633 “ cima 5636 ⟶wf 6492 –1-1→wf1 6493 ‘cfv 6496 3c3 12209 ♯chash 14230 Word cword 14402 〈“cs3 14731 Basecbs 17083 SymGrpcsymg 19148 toCycctocyc 31955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-substr 14529 df-pfx 14559 df-csh 14677 df-s2 14737 df-s3 14738 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-tset 17152 df-efmnd 18679 df-symg 19149 df-tocyc 31956 |
This theorem is referenced by: cyc3genpmlem 32000 |
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