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Mirrors > Home > MPE Home > Th. List > reldmlmhm | Structured version Visualization version GIF version |
Description: Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
reldmlmhm | ⊢ Rel dom LMHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lmhm 18843 | . 2 ⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))}) | |
2 | 1 | reldmmpt2 6669 | 1 ⊢ Rel dom LMHom |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∀wral 2896 {crab 2900 [wsbc 3402 dom cdm 5038 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 GrpHom cghm 17480 LModclmod 18686 LMHom clmhm 18840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-oprab 6553 df-mpt2 6554 df-lmhm 18843 |
This theorem is referenced by: mendbas 36773 |
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