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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 | ⊢ 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)) |
ressply1.1 | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
ressply1sub.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ressply1sub.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ressply1sub | ⊢ (𝜑 → (𝑋(-g‘𝑈)𝑌) = (𝑋(-g‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressply.1 | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
2 | ressply.2 | . . . . 5 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | ressply.3 | . . . . 5 ⊢ 𝑈 = (Poly1‘𝐻) | |
4 | ressply.4 | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
5 | ressply.5 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
6 | ressply1.1 | . . . . 5 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
7 | ressply1sub.2 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ressply1invg 33128 | . . . 4 ⊢ (𝜑 → ((invg‘𝑈)‘𝑌) = ((invg‘𝑃)‘𝑌)) |
9 | 8 | oveq2d 7418 | . . 3 ⊢ (𝜑 → (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌)) = (𝑋(+g‘𝑈)((invg‘𝑃)‘𝑌))) |
10 | ressply1sub.1 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2, 3, 4 | subrgply1 22076 | . . . . . . . . 9 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆)) |
12 | subrgsubg 20471 | . . . . . . . . 9 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆)) | |
13 | 5, 11, 12 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝑆)) |
14 | 6 | subggrp 19048 | . . . . . . . 8 ⊢ (𝐵 ∈ (SubGrp‘𝑆) → 𝑃 ∈ Grp) |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Grp) |
16 | 1, 2, 3, 4, 5, 6 | ressply1bas 22072 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
17 | 7, 16 | eleqtrd 2827 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑃)) |
18 | eqid 2724 | . . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
19 | eqid 2724 | . . . . . . . 8 ⊢ (invg‘𝑃) = (invg‘𝑃) | |
20 | 18, 19 | grpinvcl 18909 | . . . . . . 7 ⊢ ((𝑃 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑃)) → ((invg‘𝑃)‘𝑌) ∈ (Base‘𝑃)) |
21 | 15, 17, 20 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → ((invg‘𝑃)‘𝑌) ∈ (Base‘𝑃)) |
22 | 21, 16 | eleqtrrd 2828 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑃)‘𝑌) ∈ 𝐵) |
23 | 10, 22 | jca 511 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ((invg‘𝑃)‘𝑌) ∈ 𝐵)) |
24 | 1, 2, 3, 4, 5, 6 | ressply1add 22073 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ ((invg‘𝑃)‘𝑌) ∈ 𝐵)) → (𝑋(+g‘𝑈)((invg‘𝑃)‘𝑌)) = (𝑋(+g‘𝑃)((invg‘𝑃)‘𝑌))) |
25 | 23, 24 | mpdan 684 | . . 3 ⊢ (𝜑 → (𝑋(+g‘𝑈)((invg‘𝑃)‘𝑌)) = (𝑋(+g‘𝑃)((invg‘𝑃)‘𝑌))) |
26 | 9, 25 | eqtrd 2764 | . 2 ⊢ (𝜑 → (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌)) = (𝑋(+g‘𝑃)((invg‘𝑃)‘𝑌))) |
27 | eqid 2724 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
28 | eqid 2724 | . . . 4 ⊢ (invg‘𝑈) = (invg‘𝑈) | |
29 | eqid 2724 | . . . 4 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
30 | 4, 27, 28, 29 | grpsubval 18907 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(-g‘𝑈)𝑌) = (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) |
31 | 10, 7, 30 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑋(-g‘𝑈)𝑌) = (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) |
32 | 10, 16 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃)) |
33 | eqid 2724 | . . . 4 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
34 | eqid 2724 | . . . 4 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
35 | 18, 33, 19, 34 | grpsubval 18907 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝑃) ∧ 𝑌 ∈ (Base‘𝑃)) → (𝑋(-g‘𝑃)𝑌) = (𝑋(+g‘𝑃)((invg‘𝑃)‘𝑌))) |
36 | 32, 17, 35 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑋(-g‘𝑃)𝑌) = (𝑋(+g‘𝑃)((invg‘𝑃)‘𝑌))) |
37 | 26, 31, 36 | 3eqtr4d 2774 | 1 ⊢ (𝜑 → (𝑋(-g‘𝑈)𝑌) = (𝑋(-g‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 ↾s cress 17174 +gcplusg 17198 Grpcgrp 18855 invgcminusg 18856 -gcsg 18857 SubGrpcsubg 19039 SubRingcsubrg 20461 Poly1cpl1 22021 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-fzo 13626 df-seq 13965 df-hash 14289 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-hom 17222 df-cco 17223 df-0g 17388 df-gsum 17389 df-prds 17394 df-pws 17396 df-mre 17531 df-mrc 17532 df-acs 17534 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-ghm 19131 df-cntz 19225 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-subrng 20438 df-subrg 20463 df-lmod 20700 df-lss 20771 df-ascl 21720 df-psr 21773 df-mpl 21775 df-opsr 21777 df-psr1 22024 df-ply1 22026 |
This theorem is referenced by: irngss 33234 |
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