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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 32832. (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 3956 | . . 3 β’ (π β πΌ β π·) |
5 | ssrab2 4073 | . . . . . . . 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 2727 | . . . . . . . . . . . 12 β’ (Baseβπ) = (Baseβπ) | |
9 | 1, 7, 8 | tocycf 32816 | . . . . . . . . . . 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 2830 | . . . . . . . 8 β’ (π β π β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
13 | 5, 12 | sselid 3976 | . . . . . . 7 β’ (π β π β Word π·) |
14 | id 22 | . . . . . . . . . 10 β’ (π€ = π β π€ = π) | |
15 | dmeq 5900 | . . . . . . . . . 10 β’ (π€ = π β dom π€ = dom π) | |
16 | eqidd 2728 | . . . . . . . . . 10 β’ (π€ = π β π· = π·) | |
17 | 14, 15, 16 | f1eq123d 6825 | . . . . . . . . 9 β’ (π€ = π β (π€:dom π€β1-1βπ· β π:dom πβ1-1βπ·)) |
18 | 17 | elrab3 3681 | . . . . . . . 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 476 | . . . . . . 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 583 | . . . . . 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 6724 | . . . 4 β’ (π β ran π β π·) |
24 | cycpmco2.j | . . . 4 β’ (π β π½ β ran π) | |
25 | 23, 24 | sseldd 3979 | . . 3 β’ (π β π½ β π·) |
26 | 3 | eldifbd 3957 | . . . . 5 β’ (π β Β¬ πΌ β ran π) |
27 | nelne2 3035 | . . . . 5 β’ ((π½ β ran π β§ Β¬ πΌ β ran π) β π½ β πΌ) | |
28 | 24, 26, 27 | syl2anc 583 | . . . 4 β’ (π β π½ β πΌ) |
29 | 28 | necomd 2991 | . . 3 β’ (π β πΌ β π½) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 32820 | . 2 β’ (π β ((πββ¨βπΌπ½ββ©)βπΌ) = π½) |
31 | 30 | fveq2d 6895 | 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 1534 β wcel 2099 β wne 2935 {crab 3427 β cdif 3941 β¨cotp 4632 β‘ccnv 5671 dom cdm 5672 ran crn 5673 βΆwf 6538 β1-1βwf1 6539 βcfv 6542 (class class class)co 7414 1c1 11131 + caddc 11133 Word cword 14488 β¨βcs1 14569 splice csplice 14723 β¨βcs2 14816 Basecbs 17171 SymGrpcsymg 19312 toCycctocyc 32805 |
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 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
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-tp 4629 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 7865 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 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-hash 14314 df-word 14489 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-csh 14763 df-s2 14823 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-tset 17243 df-efmnd 18812 df-symg 19313 df-tocyc 32806 |
This theorem is referenced by: cycpmco2lem4 32828 |
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