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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 32031. (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 3923 | . . 3 β’ (π β πΌ β π·) |
5 | ssrab2 4038 | . . . . . . . 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 2733 | . . . . . . . . . . . 12 β’ (Baseβπ) = (Baseβπ) | |
9 | 1, 7, 8 | tocycf 32015 | . . . . . . . . . . 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 6680 | . . . . . . . . 9 β’ (π β dom π = {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
12 | 6, 11 | eleqtrd 2836 | . . . . . . . 8 β’ (π β π β {π€ β Word π· β£ π€:dom π€β1-1βπ·}) |
13 | 5, 12 | sselid 3943 | . . . . . . 7 β’ (π β π β Word π·) |
14 | id 22 | . . . . . . . . . 10 β’ (π€ = π β π€ = π) | |
15 | dmeq 5860 | . . . . . . . . . 10 β’ (π€ = π β dom π€ = dom π) | |
16 | eqidd 2734 | . . . . . . . . . 10 β’ (π€ = π β π· = π·) | |
17 | 14, 15, 16 | f1eq123d 6777 | . . . . . . . . 9 β’ (π€ = π β (π€:dom π€β1-1βπ· β π:dom πβ1-1βπ·)) |
18 | 17 | elrab3 3647 | . . . . . . . 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 478 | . . . . . . 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 585 | . . . . . 6 β’ (π β π:dom πβ1-1βπ·) |
21 | f1f 6739 | . . . . . 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 6677 | . . . 4 β’ (π β ran π β π·) |
24 | cycpmco2.j | . . . 4 β’ (π β π½ β ran π) | |
25 | 23, 24 | sseldd 3946 | . . 3 β’ (π β π½ β π·) |
26 | 3 | eldifbd 3924 | . . . . 5 β’ (π β Β¬ πΌ β ran π) |
27 | nelne2 3039 | . . . . 5 β’ ((π½ β ran π β§ Β¬ πΌ β ran π) β π½ β πΌ) | |
28 | 24, 26, 27 | syl2anc 585 | . . . 4 β’ (π β π½ β πΌ) |
29 | 28 | necomd 2996 | . . 3 β’ (π β πΌ β π½) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 32019 | . 2 β’ (π β ((πββ¨βπΌπ½ββ©)βπΌ) = π½) |
31 | 30 | fveq2d 6847 | 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 1542 β wcel 2107 β wne 2940 {crab 3406 β cdif 3908 β¨cotp 4595 β‘ccnv 5633 dom cdm 5634 ran crn 5635 βΆwf 6493 β1-1βwf1 6494 βcfv 6497 (class class class)co 7358 1c1 11057 + caddc 11059 Word cword 14408 β¨βcs1 14489 splice csplice 14643 β¨βcs2 14736 Basecbs 17088 SymGrpcsymg 19153 toCycctocyc 32004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-fl 13703 df-mod 13781 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-csh 14683 df-s2 14743 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-tset 17157 df-efmnd 18684 df-symg 19154 df-tocyc 32005 |
This theorem is referenced by: cycpmco2lem4 32027 |
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