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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngocl | Structured version Visualization version GIF version |
Description: Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringi.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringi.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
ringi.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngocl | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringi.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | ringi.2 | . . 3 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | ringi.3 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | rngosm 35744 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋) |
5 | fovrn 7356 | . 2 ⊢ ((𝐻:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) | |
6 | 4, 5 | syl3an1 1165 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 × cxp 5534 ran crn 5537 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 RingOpscrngo 35738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-1st 7739 df-2nd 7740 df-rngo 35739 |
This theorem is referenced by: rngolz 35766 rngorz 35767 rngonegmn1l 35785 rngonegmn1r 35786 rngoneglmul 35787 rngonegrmul 35788 rngosubdi 35789 rngosubdir 35790 isdrngo2 35802 rngohomco 35818 rngoisocnv 35825 crngm4 35847 rngoidl 35868 keridl 35876 prnc 35911 ispridlc 35914 pridlc3 35917 dmncan1 35920 |
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