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Mirrors > Home > MPE Home > Th. List > lmhmf | Structured version Visualization version GIF version |
Description: A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmhmf.b | ⊢ 𝐵 = (Base‘𝑆) |
lmhmf.c | ⊢ 𝐶 = (Base‘𝑇) |
Ref | Expression |
---|---|
lmhmf | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmghm 19805 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
2 | lmhmf.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
3 | lmhmf.c | . . 3 ⊢ 𝐶 = (Base‘𝑇) | |
4 | 2, 3 | ghmf 18364 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶𝐶) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 GrpHom cghm 18357 LMHom clmhm 19793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-ghm 18358 df-lmhm 19796 |
This theorem is referenced by: islmhm2 19812 lmhmco 19817 lmhmplusg 19818 lmhmvsca 19819 lmhmf1o 19820 lmhmima 19821 lmhmpreima 19822 lmhmlsp 19823 lmhmrnlss 19824 lmhmeql 19829 lspextmo 19830 lmimcnv 19841 ipcl 20779 frlmup3 20946 nmoleub2lem 23720 nmoleub2lem3 23721 nmoleub3 23725 nmhmcn 23726 dimkerim 31025 kercvrlsm 39690 lmhmfgima 39691 lnmepi 39692 lmhmfgsplit 39693 pwssplit4 39696 mendring 39799 mendlmod 39800 mendassa 39801 |
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