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Mirrors > Home > MPE Home > Th. List > srg0cl | Structured version Visualization version GIF version |
Description: The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srg0cl.b | ⊢ 𝐵 = (Base‘𝑅) |
srg0cl.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
srg0cl | ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgmnd 18332 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
2 | srg0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | srg0cl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | mndidcl 17131 | . 2 ⊢ (𝑅 ∈ Mnd → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 0gc0g 15923 Mndcmnd 17117 SRingcsrg 18328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cmn 18018 df-srg 18329 |
This theorem is referenced by: srgisid 18351 srgen1zr 18353 srglmhm 18358 srgrmhm 18359 slmd0cl 29102 slmdvs0 29109 |
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