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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 19646 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
3 | lmodsn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
4 | 3 | grpbn0 18135 | . 2 ⊢ (𝐹 ∈ Grp → 𝐵 ≠ ∅) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∅c0 4294 ‘cfv 6358 Basecbs 16486 Scalarcsca 16571 Grpcgrp 18106 LModclmod 19637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-riota 7117 df-ov 7162 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-ring 19302 df-lmod 19639 |
This theorem is referenced by: lindsrng01 44530 |
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