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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 2733 | . . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
2 | eqid 2733 | . . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
3 | 1, 2 | lmhmlem 20633 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆)))) |
4 | 3 | simplld 767 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7406 Scalarcsca 17197 GrpHom cghm 19084 LModclmod 20464 LMHom clmhm 20623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6493 df-fun 6543 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-lmhm 20626 |
This theorem is referenced by: islmhm2 20642 lmhmco 20647 lmhmplusg 20648 lmhmvsca 20649 lmhmf1o 20650 lmhmima 20651 lmhmpreima 20652 lmhmlsp 20653 lmhmrnlss 20654 reslmhm 20656 reslmhm2 20657 reslmhm2b 20658 lmhmeql 20659 lspextmo 20660 islmim 20666 lmiclcl 20674 lmhmlvec 20713 lindfmm 21374 lindsmm 21375 lmhmclm 24595 lmhmqusker 32523 kercvrlsm 41811 lmhmfgima 41812 lmhmfgsplit 41814 lmhmlnmsplit 41815 |
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