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Mirrors > Home > ILE Home > Th. List > srg0cl | 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 12963 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
2 | srg0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | srg0cl.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | mndidcl 12710 | . 2 ⊢ (𝑅 ∈ Mnd → 0 ∈ 𝐵) |
5 | 1, 4 | syl 14 | 1 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 Basecbs 12432 0gc0g 12640 Mndcmnd 12696 SRingcsrg 12959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-3 8955 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-mulr 12519 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-cmn 12904 df-srg 12960 |
This theorem is referenced by: srgisid 12982 srgen1zr 12984 srglmhm 12989 srgrmhm 12990 |
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