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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmd0cl | Structured version Visualization version GIF version |
Description: The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmd0cl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmd0cl.k | ⊢ 𝐾 = (Base‘𝐹) |
slmd0cl.z | ⊢ 0 = (0g‘𝐹) |
Ref | Expression |
---|---|
slmd0cl | ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmd0cl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | slmdsrg 30457 | . 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | slmd0cl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | slmd0cl.z | . . 3 ⊢ 0 = (0g‘𝐹) | |
5 | 3, 4 | srg0cl 18982 | . 2 ⊢ (𝐹 ∈ SRing → 0 ∈ 𝐾) |
6 | 2, 5 | syl 17 | 1 ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 ‘cfv 6182 Basecbs 16329 Scalarcsca 16414 0gc0g 16559 SRingcsrg 18968 SLModcslmd 30450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-iota 6146 df-fun 6184 df-fv 6190 df-riota 6931 df-ov 6973 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-cmn 18658 df-srg 18969 df-slmd 30451 |
This theorem is referenced by: slmd0vs 30474 |
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