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Mirrors > Home > MPE Home > Th. List > rng0cl | Structured version Visualization version GIF version |
Description: The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.) |
Ref | Expression |
---|---|
rng0cl.b | ⊢ 𝐵 = (Base‘𝑅) |
rng0cl.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
rng0cl | ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnggrp 20092 | . 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
2 | rng0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rng0cl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 18916 | . 2 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 Basecbs 17174 0gc0g 17415 Grpcgrp 18884 Rngcrng 20086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-riota 7371 df-ov 7418 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-abl 19732 df-rng 20087 |
This theorem is referenced by: rngrz 20100 cntzsubrng 20498 rnglidl0 21119 rngqiprngimf1lem 21178 |
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