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| Mirrors > Home > MPE Home > Th. List > ringcld | Structured version Visualization version GIF version | ||
| Description: Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.) | 
| Ref | Expression | 
|---|---|
| ringcld.b | ⊢ 𝐵 = (Base‘𝑅) | 
| ringcld.t | ⊢ · = (.r‘𝑅) | 
| ringcld.r | ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| ringcld.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| ringcld.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| ringcld | ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ringcld.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 2 | ringcld.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | ringcld.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 4 | ringcld.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | ringcld.t | . . 3 ⊢ · = (.r‘𝑅) | |
| 6 | 4, 5 | ringcl 20248 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) | 
| 7 | 1, 2, 3, 6 | syl3anc 1372 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 .rcmulr 17299 Ringcrg 20231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mgp 20139 df-ring 20233 | 
| This theorem is referenced by: gsumdixp 20317 xpsring1d 20331 rhmqusnsg 21296 rngqiprnglin 21313 frlmphl 21802 assa2ass 21884 assa2ass2 21885 assapropd 21893 rhmpsrlem2 21962 psrass1 21985 psrdi 21986 psrass23l 21988 psrass23 21990 mhpmulcl 22154 psdmul 22171 evls1fpws 22374 evls1muld 22377 evls1maprhm 22381 rhmcomulmpl 22387 rhmmpl 22388 mamuass 22407 mamuvs1 22410 mamuvs2 22411 mavmulass 22556 mdetrsca 22610 r1pid2 26202 elrgspnlem2 33248 elrgspnsubrunlem1 33252 erlbr2d 33269 erler 33270 rlocaddval 33273 rlocmulval 33274 rloccring 33275 rlocf1 33278 rrgsubm 33288 fracerl 33309 fracfld 33311 dvdsruasso 33414 rhmquskerlem 33454 elrspunsn 33458 ssdifidlprm 33487 mxidlirredi 33500 qsdrngilem 33523 rprmasso2 33555 unitmulrprm 33557 rprmirredlem 33559 1arithidomlem1 33564 1arithidomlem2 33565 1arithidom 33566 1arithufdlem2 33574 1arithufdlem3 33575 evl1deg1 33602 evl1deg2 33603 evl1deg3 33604 ply1dg1rt 33605 ply1mulrtss 33607 q1pdir 33624 q1pvsca 33625 r1pvsca 33626 r1pcyc 33628 r1padd1 33629 r1pid2OLD 33630 assalactf1o 33687 fldextrspunlsplem 33724 fldextrspunlsp 33725 irredminply 33758 rtelextdg2lem 33768 ply1divalg3 35648 r1peuqusdeg1 35649 aks6d1c1p4 42113 drnginvmuld 42542 rhmcomulpsr 42566 rhmpsr 42567 evlsvvval 42578 evlsbagval 42581 evlsmaprhm 42585 evlmulval 42591 selvvvval 42600 evlselv 42602 selvmul 42604 evlsmhpvvval 42610 mhphf 42612 prjspertr 42620 prjspner1 42641 | 
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