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Mirrors > Home > MPE Home > Th. List > islmhm3 | Structured version Visualization version GIF version |
Description: Property of a module homomorphism, similar to ismhm 17961. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
islmhm.k | ⊢ 𝐾 = (Scalar‘𝑆) |
islmhm.l | ⊢ 𝐿 = (Scalar‘𝑇) |
islmhm.b | ⊢ 𝐵 = (Base‘𝐾) |
islmhm.e | ⊢ 𝐸 = (Base‘𝑆) |
islmhm.m | ⊢ · = ( ·𝑠 ‘𝑆) |
islmhm.n | ⊢ × = ( ·𝑠 ‘𝑇) |
Ref | Expression |
---|---|
islmhm3 | ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islmhm.k | . . 3 ⊢ 𝐾 = (Scalar‘𝑆) | |
2 | islmhm.l | . . 3 ⊢ 𝐿 = (Scalar‘𝑇) | |
3 | islmhm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
4 | islmhm.e | . . 3 ⊢ 𝐸 = (Base‘𝑆) | |
5 | islmhm.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
6 | islmhm.n | . . 3 ⊢ × = ( ·𝑠 ‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | islmhm 19802 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
8 | 7 | baib 538 | 1 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Scalarcsca 16571 ·𝑠 cvsca 16572 GrpHom cghm 18358 LModclmod 19637 LMHom clmhm 19794 |
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-ral 3146 df-rex 3147 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-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-ov 7162 df-oprab 7163 df-mpo 7164 df-lmhm 19797 |
This theorem is referenced by: islmhm2 19813 pj1lmhm 19875 |
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