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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngo0rid | Structured version Visualization version GIF version |
Description: The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ring0cl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ring0cl.2 | ⊢ 𝑋 = ran 𝐺 |
ring0cl.3 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
rngo0rid | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ring0cl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 36076 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ring0cl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | ring0cl.3 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
5 | 3, 4 | grporid 28887 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
6 | 2, 5 | sylan 580 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ran crn 5585 ‘cfv 6426 (class class class)co 7267 1st c1st 7818 GrpOpcgr 28859 GIdcgi 28860 RingOpscrngo 36060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-fo 6432 df-fv 6434 df-riota 7224 df-ov 7270 df-1st 7820 df-2nd 7821 df-grpo 28863 df-gid 28864 df-ablo 28915 df-rngo 36061 |
This theorem is referenced by: 0idl 36191 |
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