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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressply1sub | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same subtraction operation. (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΅) |
ressply1sub.1 | β’ (π β π β π΅) |
ressply1sub.2 | β’ (π β π β π΅) |
Ref | Expression |
---|---|
ressply1sub | β’ (π β (π(-gβπ)π) = (π(-gβπ)π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressply.1 | . . . . 5 β’ π = (Poly1βπ ) | |
2 | ressply.2 | . . . . 5 β’ π» = (π βΎs π) | |
3 | ressply.3 | . . . . 5 β’ π = (Poly1βπ») | |
4 | ressply.4 | . . . . 5 β’ π΅ = (Baseβπ) | |
5 | ressply.5 | . . . . 5 β’ (π β π β (SubRingβπ )) | |
6 | ressply1.1 | . . . . 5 β’ π = (π βΎs π΅) | |
7 | ressply1sub.2 | . . . . 5 β’ (π β π β π΅) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ressply1invg 33310 | . . . 4 β’ (π β ((invgβπ)βπ) = ((invgβπ)βπ)) |
9 | 8 | oveq2d 7431 | . . 3 β’ (π β (π(+gβπ)((invgβπ)βπ)) = (π(+gβπ)((invgβπ)βπ))) |
10 | ressply1sub.1 | . . . . 5 β’ (π β π β π΅) | |
11 | 1, 2, 3, 4 | subrgply1 22158 | . . . . . . . . 9 β’ (π β (SubRingβπ ) β π΅ β (SubRingβπ)) |
12 | subrgsubg 20518 | . . . . . . . . 9 β’ (π΅ β (SubRingβπ) β π΅ β (SubGrpβπ)) | |
13 | 5, 11, 12 | 3syl 18 | . . . . . . . 8 β’ (π β π΅ β (SubGrpβπ)) |
14 | 6 | subggrp 19086 | . . . . . . . 8 β’ (π΅ β (SubGrpβπ) β π β Grp) |
15 | 13, 14 | syl 17 | . . . . . . 7 β’ (π β π β Grp) |
16 | 1, 2, 3, 4, 5, 6 | ressply1bas 22154 | . . . . . . . 8 β’ (π β π΅ = (Baseβπ)) |
17 | 7, 16 | eleqtrd 2827 | . . . . . . 7 β’ (π β π β (Baseβπ)) |
18 | eqid 2725 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
19 | eqid 2725 | . . . . . . . 8 β’ (invgβπ) = (invgβπ) | |
20 | 18, 19 | grpinvcl 18946 | . . . . . . 7 β’ ((π β Grp β§ π β (Baseβπ)) β ((invgβπ)βπ) β (Baseβπ)) |
21 | 15, 17, 20 | syl2anc 582 | . . . . . 6 β’ (π β ((invgβπ)βπ) β (Baseβπ)) |
22 | 21, 16 | eleqtrrd 2828 | . . . . 5 β’ (π β ((invgβπ)βπ) β π΅) |
23 | 10, 22 | jca 510 | . . . 4 β’ (π β (π β π΅ β§ ((invgβπ)βπ) β π΅)) |
24 | 1, 2, 3, 4, 5, 6 | ressply1add 22155 | . . . 4 β’ ((π β§ (π β π΅ β§ ((invgβπ)βπ) β π΅)) β (π(+gβπ)((invgβπ)βπ)) = (π(+gβπ)((invgβπ)βπ))) |
25 | 23, 24 | mpdan 685 | . . 3 β’ (π β (π(+gβπ)((invgβπ)βπ)) = (π(+gβπ)((invgβπ)βπ))) |
26 | 9, 25 | eqtrd 2765 | . 2 β’ (π β (π(+gβπ)((invgβπ)βπ)) = (π(+gβπ)((invgβπ)βπ))) |
27 | eqid 2725 | . . . 4 β’ (+gβπ) = (+gβπ) | |
28 | eqid 2725 | . . . 4 β’ (invgβπ) = (invgβπ) | |
29 | eqid 2725 | . . . 4 β’ (-gβπ) = (-gβπ) | |
30 | 4, 27, 28, 29 | grpsubval 18944 | . . 3 β’ ((π β π΅ β§ π β π΅) β (π(-gβπ)π) = (π(+gβπ)((invgβπ)βπ))) |
31 | 10, 7, 30 | syl2anc 582 | . 2 β’ (π β (π(-gβπ)π) = (π(+gβπ)((invgβπ)βπ))) |
32 | 10, 16 | eleqtrd 2827 | . . 3 β’ (π β π β (Baseβπ)) |
33 | eqid 2725 | . . . 4 β’ (+gβπ) = (+gβπ) | |
34 | eqid 2725 | . . . 4 β’ (-gβπ) = (-gβπ) | |
35 | 18, 33, 19, 34 | grpsubval 18944 | . . 3 β’ ((π β (Baseβπ) β§ π β (Baseβπ)) β (π(-gβπ)π) = (π(+gβπ)((invgβπ)βπ))) |
36 | 32, 17, 35 | syl2anc 582 | . 2 β’ (π β (π(-gβπ)π) = (π(+gβπ)((invgβπ)βπ))) |
37 | 26, 31, 36 | 3eqtr4d 2775 | 1 β’ (π β (π(-gβπ)π) = (π(-gβπ)π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6542 (class class class)co 7415 Basecbs 17177 βΎs cress 17206 +gcplusg 17230 Grpcgrp 18892 invgcminusg 18893 -gcsg 18894 SubGrpcsubg 19077 SubRingcsubrg 20508 Poly1cpl1 22102 |
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-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 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-se 5628 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-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-ofr 7682 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 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-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-ascl 21791 df-psr 21844 df-mpl 21846 df-opsr 21848 df-psr1 22105 df-ply1 22107 |
This theorem is referenced by: evls1subd 33312 irngss 33421 |
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