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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem1 | Structured version Visualization version GIF version |
Description: Lemma for cycpmco2 32894. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
Ref | Expression |
---|---|
cycpmco2.c | β’ π = (toCycβπ·) |
cycpmco2.s | β’ π = (SymGrpβπ·) |
cycpmco2.d | β’ (π β π· β π) |
cycpmco2.w | β’ (π β π β dom π) |
cycpmco2.i | β’ (π β πΌ β (π· β ran π)) |
cycpmco2.j | β’ (π β π½ β ran π) |
cycpmco2.e | β’ πΈ = ((β‘πβπ½) + 1) |
cycpmco2.1 | β’ π = (π splice β¨πΈ, πΈ, β¨βπΌββ©β©) |
Ref | Expression |
---|---|
cycpmco2lem1 | β’ (π β ((πβπ)β((πββ¨βπΌπ½ββ©)βπΌ)) = ((πβπ)βπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpmco2.c | . . 3 β’ π = (toCycβπ·) | |
2 | cycpmco2.d | . . 3 β’ (π β π· β π) | |
3 | cycpmco2.i | . . . 4 β’ (π β πΌ β (π· β ran π)) | |
4 | 3 | eldifad 3953 | . . 3 β’ (π β πΌ β π·) |
5 | ssrab2 4070 | . . . . . . . 8 β’ {π€ β Word π· β£ π€:dom π€β1-1βπ·} β Word π· | |
6 | cycpmco2.w | . . . . . . . . 9 β’ (π β π β dom π) | |
7 | cycpmco2.s | . . . . . . . . . . . 12 β’ π = (SymGrpβπ·) | |
8 | eqid 2725 | . . . . . . . . . . . 12 β’ (Baseβπ) = (Baseβπ) | |
9 | 1, 7, 8 | tocycf 32878 | . . . . . . . . . . 11 β’ (π· β π β π:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
10 | 2, 9 | syl 17 | . . . . . . . . . 10 β’ (π β π:{π€ β Word π· β£ π€:dom π€β1-1βπ·}βΆ(Baseβπ)) |
11 | 10 | fdmd 6727 | . . . . . . . . 9 β’ (π β dom π = {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
12 | 6, 11 | eleqtrd 2827 | . . . . . . . 8 β’ (π β π β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
13 | 5, 12 | sselid 3971 | . . . . . . 7 β’ (π β π β Word π·) |
14 | id 22 | . . . . . . . . . 10 β’ (π€ = π β π€ = π) | |
15 | dmeq 5901 | . . . . . . . . . 10 β’ (π€ = π β dom π€ = dom π) | |
16 | eqidd 2726 | . . . . . . . . . 10 β’ (π€ = π β π· = π·) | |
17 | 14, 15, 16 | f1eq123d 6824 | . . . . . . . . 9 β’ (π€ = π β (π€:dom π€β1-1βπ· β π:dom πβ1-1βπ·)) |
18 | 17 | elrab3 3677 | . . . . . . . 8 β’ (π β Word π· β (π β {π€ β Word π· β£ π€:dom π€β1-1βπ·} β π:dom πβ1-1βπ·)) |
19 | 18 | biimpa 475 | . . . . . . 7 β’ ((π β Word π· β§ π β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) β π:dom πβ1-1βπ·) |
20 | 13, 12, 19 | syl2anc 582 | . . . . . 6 β’ (π β π:dom πβ1-1βπ·) |
21 | f1f 6787 | . . . . . 6 β’ (π:dom πβ1-1βπ· β π:dom πβΆπ·) | |
22 | 20, 21 | syl 17 | . . . . 5 β’ (π β π:dom πβΆπ·) |
23 | 22 | frnd 6725 | . . . 4 β’ (π β ran π β π·) |
24 | cycpmco2.j | . . . 4 β’ (π β π½ β ran π) | |
25 | 23, 24 | sseldd 3974 | . . 3 β’ (π β π½ β π·) |
26 | 3 | eldifbd 3954 | . . . . 5 β’ (π β Β¬ πΌ β ran π) |
27 | nelne2 3030 | . . . . 5 β’ ((π½ β ran π β§ Β¬ πΌ β ran π) β π½ β πΌ) | |
28 | 24, 26, 27 | syl2anc 582 | . . . 4 β’ (π β π½ β πΌ) |
29 | 28 | necomd 2986 | . . 3 β’ (π β πΌ β π½) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 32882 | . 2 β’ (π β ((πββ¨βπΌπ½ββ©)βπΌ) = π½) |
31 | 30 | fveq2d 6894 | 1 β’ (π β ((πβπ)β((πββ¨βπΌπ½ββ©)βπΌ)) = ((πβπ)βπ½)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1533 β wcel 2098 β wne 2930 {crab 3419 β cdif 3938 β¨cotp 4633 β‘ccnv 5672 dom cdm 5673 ran crn 5674 βΆwf 6539 β1-1βwf1 6540 βcfv 6543 (class class class)co 7413 1c1 11134 + caddc 11136 Word cword 14491 β¨βcs1 14572 splice csplice 14726 β¨βcs2 14819 Basecbs 17174 SymGrpcsymg 19320 toCycctocyc 32867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-hash 14317 df-word 14492 df-concat 14548 df-s1 14573 df-substr 14618 df-pfx 14648 df-csh 14766 df-s2 14826 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-tset 17246 df-efmnd 18820 df-symg 19321 df-tocyc 32868 |
This theorem is referenced by: cycpmco2lem4 32890 |
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