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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 2737 | . . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
2 | eqid 2737 | . . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
3 | 1, 2 | lmhmlem 20066 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆)))) |
4 | 3 | simplld 768 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Scalarcsca 16805 GrpHom cghm 18619 LModclmod 19899 LMHom clmhm 20056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-lmhm 20059 |
This theorem is referenced by: islmhm2 20075 lmhmco 20080 lmhmplusg 20081 lmhmvsca 20082 lmhmf1o 20083 lmhmima 20084 lmhmpreima 20085 lmhmlsp 20086 lmhmrnlss 20087 reslmhm 20089 reslmhm2 20090 reslmhm2b 20091 lmhmeql 20092 lspextmo 20093 islmim 20099 lmiclcl 20107 lindfmm 20789 lindsmm 20790 lmhmclm 23984 lmhmlvec 39973 kercvrlsm 40611 lmhmfgima 40612 lmhmfgsplit 40614 lmhmlnmsplit 40615 |
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