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Mirrors > Home > MPE Home > Th. List > rngmgp | Structured version Visualization version GIF version |
Description: A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.) |
Ref | Expression |
---|---|
rngmgp.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
rngmgp | ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | rngmgp.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | eqid 2728 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2728 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isrng 20101 | . 2 ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ Smgrp ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))))) |
6 | 5 | simp2bi 1143 | 1 ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 Smgrpcsgrp 18685 Abelcabl 19743 mulGrpcmgp 20081 Rngcrng 20099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-rng 20100 |
This theorem is referenced by: rngmgpf 20104 rngass 20106 rngcl 20111 isringrng 20230 isrnghmmul 20388 idrnghm 20404 c0rnghm 20479 cntzsubrng 20511 rnglidlmsgrp 21148 |
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