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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 35182 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋) |
5 | fovrn 7321 | . 2 ⊢ ((𝐻:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) | |
6 | 4, 5 | syl3an1 1159 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 × cxp 5556 ran crn 5559 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 1st c1st 7690 2nd c2nd 7691 RingOpscrngo 35176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-1st 7692 df-2nd 7693 df-rngo 35177 |
This theorem is referenced by: rngolz 35204 rngorz 35205 rngonegmn1l 35223 rngonegmn1r 35224 rngoneglmul 35225 rngonegrmul 35226 rngosubdi 35227 rngosubdir 35228 isdrngo2 35240 rngohomco 35256 rngoisocnv 35263 crngm4 35285 rngoidl 35306 keridl 35314 prnc 35349 ispridlc 35352 pridlc3 35355 dmncan1 35358 |
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