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| Mirrors > Home > MPE Home > Th. List > islmhmd | Structured version Visualization version GIF version | ||
| Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| islmhmd.x | ⊢ 𝑋 = (Base‘𝑆) | 
| islmhmd.a | ⊢ · = ( ·𝑠 ‘𝑆) | 
| islmhmd.b | ⊢ × = ( ·𝑠 ‘𝑇) | 
| islmhmd.k | ⊢ 𝐾 = (Scalar‘𝑆) | 
| islmhmd.j | ⊢ 𝐽 = (Scalar‘𝑇) | 
| islmhmd.n | ⊢ 𝑁 = (Base‘𝐾) | 
| islmhmd.s | ⊢ (𝜑 → 𝑆 ∈ LMod) | 
| islmhmd.t | ⊢ (𝜑 → 𝑇 ∈ LMod) | 
| islmhmd.c | ⊢ (𝜑 → 𝐽 = 𝐾) | 
| islmhmd.f | ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | 
| islmhmd.l | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) | 
| Ref | Expression | 
|---|---|
| islmhmd | ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | islmhmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ LMod) | |
| 2 | islmhmd.t | . 2 ⊢ (𝜑 → 𝑇 ∈ LMod) | |
| 3 | islmhmd.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
| 4 | islmhmd.c | . . 3 ⊢ (𝜑 → 𝐽 = 𝐾) | |
| 5 | islmhmd.l | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) | |
| 6 | 5 | ralrimivva 3201 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) | 
| 7 | 3, 4, 6 | 3jca 1128 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))) | 
| 8 | islmhmd.k | . . 3 ⊢ 𝐾 = (Scalar‘𝑆) | |
| 9 | islmhmd.j | . . 3 ⊢ 𝐽 = (Scalar‘𝑇) | |
| 10 | islmhmd.n | . . 3 ⊢ 𝑁 = (Base‘𝐾) | |
| 11 | islmhmd.x | . . 3 ⊢ 𝑋 = (Base‘𝑆) | |
| 12 | islmhmd.a | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
| 13 | islmhmd.b | . . 3 ⊢ × = ( ·𝑠 ‘𝑇) | |
| 14 | 8, 9, 10, 11, 12, 13 | islmhm 21027 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) | 
| 15 | 1, 2, 7, 14 | syl21anbrc 1344 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Scalarcsca 17301 ·𝑠 cvsca 17302 GrpHom cghm 19231 LModclmod 20859 LMHom clmhm 21019 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-lmhm 21022 | 
| This theorem is referenced by: 0lmhm 21040 idlmhm 21041 invlmhm 21042 lmhmco 21043 lmhmplusg 21044 lmhmvsca 21045 lmhmf1o 21046 reslmhm2 21053 reslmhm2b 21054 pwsdiaglmhm 21057 pwssplit3 21061 frlmup1 21819 imaslmhm 33386 quslmhm 33388 lmhmqusker 33446 frlmsnic 42555 | 
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