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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdsrg | Structured version Visualization version Unicode version |
Description: The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdsrg.1 | Scalar |
Ref | Expression |
---|---|
slmdsrg | SLMod SRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 | |
2 | eqid 2622 | . . 3 | |
3 | eqid 2622 | . . 3 | |
4 | eqid 2622 | . . 3 | |
5 | slmdsrg.1 | . . 3 Scalar | |
6 | eqid 2622 | . . 3 | |
7 | eqid 2622 | . . 3 | |
8 | eqid 2622 | . . 3 | |
9 | eqid 2622 | . . 3 | |
10 | eqid 2622 | . . 3 | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 29755 | . 2 SLMod CMnd SRing |
12 | 11 | simp2bi 1077 | 1 SLMod SRing |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmulr 15942 Scalarcsca 15944 cvsca 15945 c0g 16100 CMndccmn 18193 cur 18501 SRingcsrg 18505 SLModcslmd 29753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-slmd 29754 |
This theorem is referenced by: slmdacl 29762 slmdmcl 29763 slmdsn0 29764 slmd0cl 29771 slmd1cl 29772 slmdvs0 29778 gsumvsca2 29783 |
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