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Mirrors > Home > MPE Home > Th. List > Mathboxes > mplringd | Structured version Visualization version GIF version |
Description: The polynomial ring is a ring. (Contributed by SN, 7-Feb-2025.) |
Ref | Expression |
---|---|
mplringd.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplringd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mplringd.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
mplringd | ⊢ (𝜑 → 𝑃 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplringd.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
2 | mplringd.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
3 | mplringd.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | 3 | mplring 21503 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) |
5 | 1, 2, 4 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑃 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 (class class class)co 7392 Ringcrg 20013 mPoly cmpl 21387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-iin 4992 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-se 5624 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7652 df-ofr 7653 df-om 7838 df-1st 7956 df-2nd 7957 df-supp 8128 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-1o 8447 df-er 8685 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9344 df-sup 9418 df-oi 9486 df-card 9915 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12454 df-z 12540 df-dec 12659 df-uz 12804 df-fz 13466 df-fzo 13609 df-seq 13948 df-hash 14272 df-struct 17061 df-sets 17078 df-slot 17096 df-ndx 17108 df-base 17126 df-ress 17155 df-plusg 17191 df-mulr 17192 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-hom 17202 df-cco 17203 df-0g 17368 df-gsum 17369 df-prds 17374 df-pws 17376 df-mre 17511 df-mrc 17512 df-acs 17514 df-mgm 18542 df-sgrp 18591 df-mnd 18602 df-mhm 18646 df-submnd 18647 df-grp 18796 df-minusg 18797 df-mulg 18922 df-subg 18974 df-ghm 19055 df-cntz 19146 df-cmn 19613 df-abl 19614 df-mgp 19946 df-ur 19963 df-ring 20015 df-subrg 20307 df-psr 21390 df-mpl 21392 |
This theorem is referenced by: rhmmpl 40915 |
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