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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 20268 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
7 | 1, 2, 3, 6 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Ringcrg 20251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mgp 20153 df-ring 20253 |
This theorem is referenced by: gsumdixp 20333 xpsring1d 20347 rhmqusnsg 21313 rngqiprnglin 21330 frlmphl 21819 assa2ass 21901 assa2ass2 21902 assapropd 21910 rhmpsrlem2 21979 psrass1 22002 psrdi 22003 psrass23l 22005 psrass23 22007 mhpmulcl 22171 psdmul 22188 evls1fpws 22389 evls1muld 22392 evls1maprhm 22396 rhmcomulmpl 22402 rhmmpl 22403 mamuass 22422 mamuvs1 22425 mamuvs2 22426 mavmulass 22571 mdetrsca 22625 r1pid2 26216 elrgspnlem2 33233 erlbr2d 33251 erler 33252 rlocaddval 33255 rlocmulval 33256 rloccring 33257 rlocf1 33260 rrgsubm 33268 fracerl 33288 fracfld 33290 dvdsruasso 33393 rhmquskerlem 33433 elrspunsn 33437 ssdifidlprm 33466 mxidlirredi 33479 qsdrngilem 33502 rprmasso2 33534 unitmulrprm 33536 rprmirredlem 33538 1arithidomlem1 33543 1arithidomlem2 33544 1arithidom 33545 1arithufdlem2 33553 1arithufdlem3 33554 evl1deg1 33581 evl1deg2 33582 evl1deg3 33583 ply1dg1rt 33584 ply1mulrtss 33586 q1pdir 33603 q1pvsca 33604 r1pvsca 33605 r1pcyc 33607 r1padd1 33608 r1pid2OLD 33609 assalactf1o 33663 irredminply 33722 rtelextdg2lem 33732 ply1divalg3 35627 r1peuqusdeg1 35628 aks6d1c1p4 42093 drnginvmuld 42514 rhmcomulpsr 42538 rhmpsr 42539 evlsvvval 42550 evlsbagval 42553 evlsmaprhm 42557 evlmulval 42563 selvvvval 42572 evlselv 42574 selvmul 42576 evlsmhpvvval 42582 mhphf 42584 prjspertr 42592 prjspner1 42613 |
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