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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 19261 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
2 | srg0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | srg0cl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | mndidcl 17928 | . 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 1537 ∈ wcel 2114 ‘cfv 6357 Basecbs 16485 0gc0g 16715 Mndcmnd 17913 SRingcsrg 19257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 df-ov 7161 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-cmn 18910 df-srg 19258 |
This theorem is referenced by: srgisid 19280 srgen1zr 19282 srglmhm 19287 srgrmhm 19288 slmd0cl 30848 slmdvs0 30855 |
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