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Mirrors > Home > MPE Home > Th. List > lmodsn0 | Structured version Visualization version GIF version |
Description: The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodsn0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodsn0.b | ⊢ 𝐵 = (Base‘𝐹) |
Ref | Expression |
---|---|
lmodsn0 | ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodsn0.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | lmodfgrp 18872 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
3 | lmodsn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
4 | 3 | grpbn0 17451 | . 2 ⊢ (𝐹 ∈ Grp → 𝐵 ≠ ∅) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 ‘cfv 5888 Basecbs 15857 Scalarcsca 15944 Grpcgrp 17422 LModclmod 18863 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
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-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ring 18549 df-lmod 18865 |
This theorem is referenced by: lindsrng01 42257 |
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