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Mirrors > Home > MPE Home > Th. List > frmdsssubm | Structured version Visualization version GIF version |
Description: The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
frmdmnd.m | β’ π = (freeMndβπΌ) |
Ref | Expression |
---|---|
frmdsssubm | β’ ((πΌ β π β§ π½ β πΌ) β Word π½ β (SubMndβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sswrd 14502 | . . . 4 β’ (π½ β πΌ β Word π½ β Word πΌ) | |
2 | 1 | adantl 480 | . . 3 β’ ((πΌ β π β§ π½ β πΌ) β Word π½ β Word πΌ) |
3 | frmdmnd.m | . . . . 5 β’ π = (freeMndβπΌ) | |
4 | eqid 2725 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
5 | 3, 4 | frmdbas 18806 | . . . 4 β’ (πΌ β π β (Baseβπ) = Word πΌ) |
6 | 5 | adantr 479 | . . 3 β’ ((πΌ β π β§ π½ β πΌ) β (Baseβπ) = Word πΌ) |
7 | 2, 6 | sseqtrrd 4014 | . 2 β’ ((πΌ β π β§ π½ β πΌ) β Word π½ β (Baseβπ)) |
8 | wrd0 14519 | . . 3 β’ β β Word π½ | |
9 | 8 | a1i 11 | . 2 β’ ((πΌ β π β§ π½ β πΌ) β β β Word π½) |
10 | 7 | sselda 3972 | . . . . . 6 β’ (((πΌ β π β§ π½ β πΌ) β§ π₯ β Word π½) β π₯ β (Baseβπ)) |
11 | 7 | sselda 3972 | . . . . . 6 β’ (((πΌ β π β§ π½ β πΌ) β§ π¦ β Word π½) β π¦ β (Baseβπ)) |
12 | 10, 11 | anim12dan 617 | . . . . 5 β’ (((πΌ β π β§ π½ β πΌ) β§ (π₯ β Word π½ β§ π¦ β Word π½)) β (π₯ β (Baseβπ) β§ π¦ β (Baseβπ))) |
13 | eqid 2725 | . . . . . 6 β’ (+gβπ) = (+gβπ) | |
14 | 3, 4, 13 | frmdadd 18809 | . . . . 5 β’ ((π₯ β (Baseβπ) β§ π¦ β (Baseβπ)) β (π₯(+gβπ)π¦) = (π₯ ++ π¦)) |
15 | 12, 14 | syl 17 | . . . 4 β’ (((πΌ β π β§ π½ β πΌ) β§ (π₯ β Word π½ β§ π¦ β Word π½)) β (π₯(+gβπ)π¦) = (π₯ ++ π¦)) |
16 | ccatcl 14554 | . . . . 5 β’ ((π₯ β Word π½ β§ π¦ β Word π½) β (π₯ ++ π¦) β Word π½) | |
17 | 16 | adantl 480 | . . . 4 β’ (((πΌ β π β§ π½ β πΌ) β§ (π₯ β Word π½ β§ π¦ β Word π½)) β (π₯ ++ π¦) β Word π½) |
18 | 15, 17 | eqeltrd 2825 | . . 3 β’ (((πΌ β π β§ π½ β πΌ) β§ (π₯ β Word π½ β§ π¦ β Word π½)) β (π₯(+gβπ)π¦) β Word π½) |
19 | 18 | ralrimivva 3191 | . 2 β’ ((πΌ β π β§ π½ β πΌ) β βπ₯ β Word π½βπ¦ β Word π½(π₯(+gβπ)π¦) β Word π½) |
20 | 3 | frmdmnd 18813 | . . . 4 β’ (πΌ β π β π β Mnd) |
21 | 20 | adantr 479 | . . 3 β’ ((πΌ β π β§ π½ β πΌ) β π β Mnd) |
22 | 3 | frmd0 18814 | . . . 4 β’ β = (0gβπ) |
23 | 4, 22, 13 | issubm 18757 | . . 3 β’ (π β Mnd β (Word π½ β (SubMndβπ) β (Word π½ β (Baseβπ) β§ β β Word π½ β§ βπ₯ β Word π½βπ¦ β Word π½(π₯(+gβπ)π¦) β Word π½))) |
24 | 21, 23 | syl 17 | . 2 β’ ((πΌ β π β§ π½ β πΌ) β (Word π½ β (SubMndβπ) β (Word π½ β (Baseβπ) β§ β β Word π½ β§ βπ₯ β Word π½βπ¦ β Word π½(π₯(+gβπ)π¦) β Word π½))) |
25 | 7, 9, 19, 24 | mpbir3and 1339 | 1 β’ ((πΌ β π β§ π½ β πΌ) β Word π½ β (SubMndβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 β wss 3940 β c0 4318 βcfv 6542 (class class class)co 7415 Word cword 14494 ++ cconcat 14550 Basecbs 17177 +gcplusg 17230 Mndcmnd 18691 SubMndcsubmnd 18736 freeMndcfrmd 18801 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-concat 14551 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-frmd 18803 |
This theorem is referenced by: (None) |
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