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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressply10g | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.) |
Ref | Expression |
---|---|
ressply.1 | ⊢ 𝑆 = (Poly1‘𝑅) |
ressply.2 | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressply.3 | ⊢ 𝑈 = (Poly1‘𝐻) |
ressply.4 | ⊢ 𝐵 = (Base‘𝑈) |
ressply.5 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressply10g.6 | ⊢ 𝑍 = (0g‘𝑆) |
Ref | Expression |
---|---|
ressply10g | ⊢ (𝜑 → 𝑍 = (0g‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressply.1 | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
2 | eqid 2733 | . . . . 5 ⊢ (algSc‘𝑆) = (algSc‘𝑆) | |
3 | ressply.2 | . . . . 5 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
4 | ressply.3 | . . . . 5 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | ressply.5 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
6 | eqid 2733 | . . . . 5 ⊢ (algSc‘𝑈) = (algSc‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | subrg1ascl 21762 | . . . 4 ⊢ (𝜑 → (algSc‘𝑈) = ((algSc‘𝑆) ↾ 𝑇)) |
8 | 7 | fveq1d 6889 | . . 3 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘𝐻)) = (((algSc‘𝑆) ↾ 𝑇)‘(0g‘𝐻))) |
9 | eqid 2733 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
10 | eqid 2733 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
11 | 3 | subrgring 20353 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
12 | 5, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ Ring) |
13 | 4, 6, 9, 10, 12 | ply1ascl0 32601 | . . 3 ⊢ (𝜑 → ((algSc‘𝑈)‘(0g‘𝐻)) = (0g‘𝑈)) |
14 | eqid 2733 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 3, 14 | subrg0 20357 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
16 | 5, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
17 | subrgsubg 20356 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅)) | |
18 | 14 | subg0cl 19007 | . . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑇) |
19 | 5, 17, 18 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝑇) |
20 | 16, 19 | eqeltrrd 2835 | . . . 4 ⊢ (𝜑 → (0g‘𝐻) ∈ 𝑇) |
21 | 20 | fvresd 6907 | . . 3 ⊢ (𝜑 → (((algSc‘𝑆) ↾ 𝑇)‘(0g‘𝐻)) = ((algSc‘𝑆)‘(0g‘𝐻))) |
22 | 8, 13, 21 | 3eqtr3d 2781 | . 2 ⊢ (𝜑 → (0g‘𝑈) = ((algSc‘𝑆)‘(0g‘𝐻))) |
23 | 16 | fveq2d 6891 | . 2 ⊢ (𝜑 → ((algSc‘𝑆)‘(0g‘𝑅)) = ((algSc‘𝑆)‘(0g‘𝐻))) |
24 | ressply10g.6 | . . 3 ⊢ 𝑍 = (0g‘𝑆) | |
25 | subrgrcl 20355 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
26 | 5, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
27 | 1, 2, 14, 24, 26 | ply1ascl0 32601 | . 2 ⊢ (𝜑 → ((algSc‘𝑆)‘(0g‘𝑅)) = 𝑍) |
28 | 22, 23, 27 | 3eqtr2rd 2780 | 1 ⊢ (𝜑 → 𝑍 = (0g‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ↾ cres 5676 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 ↾s cress 17168 0gc0g 17380 SubGrpcsubg 18993 Ringcrg 20046 SubRingcsubrg 20346 algSccascl 21390 Poly1cpl1 21682 |
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 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-fz 13480 df-fzo 13623 df-seq 13962 df-hash 14286 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-hom 17216 df-cco 17217 df-0g 17382 df-gsum 17383 df-prds 17388 df-pws 17390 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-mhm 18666 df-submnd 18667 df-grp 18817 df-minusg 18818 df-sbg 18819 df-mulg 18944 df-subg 18996 df-ghm 19083 df-cntz 19174 df-cmn 19642 df-abl 19643 df-mgp 19979 df-ur 19996 df-ring 20048 df-subrg 20348 df-lmod 20460 df-lss 20530 df-ascl 21393 df-psr 21443 df-mpl 21445 df-opsr 21447 df-psr1 21685 df-ply1 21687 |
This theorem is referenced by: ressply1mon1p 32603 ressply1invg 32604 irngnzply1lem 32698 irngnzply1 32699 |
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