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Mirrors > Home > MPE Home > Th. List > lmhmlmod1 | Structured version Visualization version GIF version |
Description: A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmhmlmod1 | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
2 | eqid 2818 | . . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
3 | 1, 2 | lmhmlem 19730 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆)))) |
4 | 3 | simplld 764 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Scalarcsca 16556 GrpHom cghm 18293 LModclmod 19563 LMHom clmhm 19720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-lmhm 19723 |
This theorem is referenced by: islmhm2 19739 lmhmco 19744 lmhmplusg 19745 lmhmvsca 19746 lmhmf1o 19747 lmhmima 19748 lmhmpreima 19749 lmhmlsp 19750 lmhmrnlss 19751 reslmhm 19753 reslmhm2 19754 reslmhm2b 19755 lmhmeql 19756 lspextmo 19757 islmim 19763 lmiclcl 19771 lindfmm 20899 lindsmm 20900 lmhmclm 23618 lmhmlvec 39026 kercvrlsm 39561 lmhmfgima 39562 lmhmfgsplit 39564 lmhmlnmsplit 39565 |
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