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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressply1invg | Structured version Visualization version GIF version |
Description: An element of a restricted polynomial algebra has the same group inverse. (Contributed by Thierry Arnoux, 30-Jan-2025.) |
Ref | Expression |
---|---|
ressply.1 | β’ π = (Poly1βπ ) |
ressply.2 | β’ π» = (π βΎs π) |
ressply.3 | β’ π = (Poly1βπ») |
ressply.4 | β’ π΅ = (Baseβπ) |
ressply.5 | β’ (π β π β (SubRingβπ )) |
ressply1.1 | β’ π = (π βΎs π΅) |
ressply1invg.1 | β’ (π β π β π΅) |
Ref | Expression |
---|---|
ressply1invg | β’ (π β ((invgβπ)βπ) = ((invgβπ)βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressply.1 | . . . 4 β’ π = (Poly1βπ ) | |
2 | ressply.2 | . . . 4 β’ π» = (π βΎs π) | |
3 | ressply.3 | . . . 4 β’ π = (Poly1βπ») | |
4 | ressply.4 | . . . 4 β’ π΅ = (Baseβπ) | |
5 | ressply.5 | . . . 4 β’ (π β π β (SubRingβπ )) | |
6 | ressply1.1 | . . . 4 β’ π = (π βΎs π΅) | |
7 | 1, 2, 3, 4, 5, 6 | ressply1bas 22151 | . . 3 β’ (π β π΅ = (Baseβπ)) |
8 | ressply1invg.1 | . . . . 5 β’ (π β π β π΅) | |
9 | 1, 2, 3, 4, 5, 6 | ressply1add 22152 | . . . . . 6 β’ ((π β§ (π¦ β π΅ β§ π β π΅)) β (π¦(+gβπ)π) = (π¦(+gβπ)π)) |
10 | 9 | anassrs 466 | . . . . 5 β’ (((π β§ π¦ β π΅) β§ π β π΅) β (π¦(+gβπ)π) = (π¦(+gβπ)π)) |
11 | 8, 10 | mpidan 687 | . . . 4 β’ ((π β§ π¦ β π΅) β (π¦(+gβπ)π) = (π¦(+gβπ)π)) |
12 | eqid 2725 | . . . . . . 7 β’ (0gβπ) = (0gβπ) | |
13 | 1, 2, 3, 4, 5, 12 | ressply10g 33305 | . . . . . 6 β’ (π β (0gβπ) = (0gβπ)) |
14 | 1, 2, 3, 4 | subrgply1 22155 | . . . . . . . . 9 β’ (π β (SubRingβπ ) β π΅ β (SubRingβπ)) |
15 | 5, 14 | syl 17 | . . . . . . . 8 β’ (π β π΅ β (SubRingβπ)) |
16 | subrgrcl 20514 | . . . . . . . 8 β’ (π΅ β (SubRingβπ) β π β Ring) | |
17 | ringmnd 20182 | . . . . . . . 8 β’ (π β Ring β π β Mnd) | |
18 | 15, 16, 17 | 3syl 18 | . . . . . . 7 β’ (π β π β Mnd) |
19 | subrgsubg 20515 | . . . . . . . 8 β’ (π΅ β (SubRingβπ) β π΅ β (SubGrpβπ)) | |
20 | 12 | subg0cl 19088 | . . . . . . . 8 β’ (π΅ β (SubGrpβπ) β (0gβπ) β π΅) |
21 | 15, 19, 20 | 3syl 18 | . . . . . . 7 β’ (π β (0gβπ) β π΅) |
22 | eqid 2725 | . . . . . . . . 9 β’ (PwSer1βπ») = (PwSer1βπ») | |
23 | eqid 2725 | . . . . . . . . 9 β’ (Baseβ(PwSer1βπ»)) = (Baseβ(PwSer1βπ»)) | |
24 | eqid 2725 | . . . . . . . . 9 β’ (Baseβπ) = (Baseβπ) | |
25 | 1, 2, 3, 4, 5, 22, 23, 24 | ressply1bas2 22150 | . . . . . . . 8 β’ (π β π΅ = ((Baseβ(PwSer1βπ»)) β© (Baseβπ))) |
26 | inss2 4225 | . . . . . . . 8 β’ ((Baseβ(PwSer1βπ»)) β© (Baseβπ)) β (Baseβπ) | |
27 | 25, 26 | eqsstrdi 4028 | . . . . . . 7 β’ (π β π΅ β (Baseβπ)) |
28 | 6, 24, 12 | ress0g 18716 | . . . . . . 7 β’ ((π β Mnd β§ (0gβπ) β π΅ β§ π΅ β (Baseβπ)) β (0gβπ) = (0gβπ)) |
29 | 18, 21, 27, 28 | syl3anc 1368 | . . . . . 6 β’ (π β (0gβπ) = (0gβπ)) |
30 | 13, 29 | eqtr3d 2767 | . . . . 5 β’ (π β (0gβπ) = (0gβπ)) |
31 | 30 | adantr 479 | . . . 4 β’ ((π β§ π¦ β π΅) β (0gβπ) = (0gβπ)) |
32 | 11, 31 | eqeq12d 2741 | . . 3 β’ ((π β§ π¦ β π΅) β ((π¦(+gβπ)π) = (0gβπ) β (π¦(+gβπ)π) = (0gβπ))) |
33 | 7, 32 | riotaeqbidva 32334 | . 2 β’ (π β (β©π¦ β π΅ (π¦(+gβπ)π) = (0gβπ)) = (β©π¦ β (Baseβπ)(π¦(+gβπ)π) = (0gβπ))) |
34 | eqid 2725 | . . . 4 β’ (+gβπ) = (+gβπ) | |
35 | eqid 2725 | . . . 4 β’ (0gβπ) = (0gβπ) | |
36 | eqid 2725 | . . . 4 β’ (invgβπ) = (invgβπ) | |
37 | 4, 34, 35, 36 | grpinvval 18936 | . . 3 β’ (π β π΅ β ((invgβπ)βπ) = (β©π¦ β π΅ (π¦(+gβπ)π) = (0gβπ))) |
38 | 8, 37 | syl 17 | . 2 β’ (π β ((invgβπ)βπ) = (β©π¦ β π΅ (π¦(+gβπ)π) = (0gβπ))) |
39 | 8, 7 | eleqtrd 2827 | . . 3 β’ (π β π β (Baseβπ)) |
40 | eqid 2725 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
41 | eqid 2725 | . . . 4 β’ (+gβπ) = (+gβπ) | |
42 | eqid 2725 | . . . 4 β’ (0gβπ) = (0gβπ) | |
43 | eqid 2725 | . . . 4 β’ (invgβπ) = (invgβπ) | |
44 | 40, 41, 42, 43 | grpinvval 18936 | . . 3 β’ (π β (Baseβπ) β ((invgβπ)βπ) = (β©π¦ β (Baseβπ)(π¦(+gβπ)π) = (0gβπ))) |
45 | 39, 44 | syl 17 | . 2 β’ (π β ((invgβπ)βπ) = (β©π¦ β (Baseβπ)(π¦(+gβπ)π) = (0gβπ))) |
46 | 33, 38, 45 | 3eqtr4d 2775 | 1 β’ (π β ((invgβπ)βπ) = ((invgβπ)βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β© cin 3940 β wss 3941 βcfv 6543 β©crio 7368 (class class class)co 7413 Basecbs 17174 βΎs cress 17203 +gcplusg 17227 0gc0g 17415 Mndcmnd 18688 invgcminusg 18890 SubGrpcsubg 19074 Ringcrg 20172 SubRingcsubrg 20505 PwSer1cps1 22097 Poly1cpl1 22099 |
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 |
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-iin 4995 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-se 5629 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-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 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-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-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-ascl 21788 df-psr 21841 df-mpl 21843 df-opsr 21845 df-psr1 22102 df-ply1 22104 |
This theorem is referenced by: ressply1sub 33308 |
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