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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 20181 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
7 | 1, 2, 3, 6 | syl3anc 1369 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 .rcmulr 17225 Ringcrg 20164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mgp 20066 df-ring 20166 |
This theorem is referenced by: gsumdixp 20244 xpsring1d 20258 rngqiprnglin 21181 frlmphl 21702 assapropd 21792 psrmulcllem 21875 psrass1 21894 psrass23l 21897 psrass23 21899 psdmul 22077 mamuass 22289 mamuvs1 22292 mamuvs2 22293 mavmulass 22438 mdetrsca 22492 dvdsruasso 33029 rhmquskerlem 33076 elrspunsn 33080 mxidlirredi 33120 qsdrngilem 33141 evls1fpws 33182 evls1muld 33186 q1pdir 33205 q1pvsca 33206 r1pvsca 33207 r1pcyc 33209 r1padd1 33210 r1pid2 33211 evls1maprhm 33305 irredminply 33320 aks6d1c1p4 41515 drnginvmuld 41685 rhmmpllem2 41705 rhmcomulmpl 41707 rhmmpl 41708 evlsvvval 41718 evlsbagval 41721 evlsmaprhm 41725 evlmulval 41731 selvvvval 41740 evlselv 41742 selvmul 41744 evlsmhpvvval 41750 mhphf 41752 prjspner1 41972 |
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