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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngo0lid | Structured version Visualization version GIF version |
Description: The additive identity of a ring is a left 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 |
---|---|
rngo0lid | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ring0cl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 35348 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ring0cl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | ring0cl.3 | . . 3 ⊢ 𝑍 = (GId‘𝐺) | |
5 | 3, 4 | grpolid 28299 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) |
6 | 2, 5 | sylan 583 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ran crn 5520 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 GrpOpcgr 28272 GIdcgi 28273 RingOpscrngo 35332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-1st 7671 df-2nd 7672 df-grpo 28276 df-gid 28277 df-ablo 28328 df-rngo 35333 |
This theorem is referenced by: (None) |
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