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Theorem lmhmplusg 20800
Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
lmhmplusg.p + = (+gβ€˜π‘)
Assertion
Ref Expression
lmhmplusg ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Proof of Theorem lmhmplusg
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . 2 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2731 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 eqid 2731 . 2 ( ·𝑠 β€˜π‘) = ( ·𝑠 β€˜π‘)
4 eqid 2731 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 eqid 2731 . 2 (Scalarβ€˜π‘) = (Scalarβ€˜π‘)
6 eqid 2731 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20789 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
87adantr 480 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20788 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
109adantr 480 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20786 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
1211adantr 480 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
13 lmodabl 20664 . . . 4 (𝑁 ∈ LMod β†’ 𝑁 ∈ Abel)
1410, 13syl 17 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ Abel)
15 lmghm 20787 . . . 4 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
1615adantr 480 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
17 lmghm 20787 . . . 4 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
1817adantl 481 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
19 lmhmplusg.p . . . 4 + = (+gβ€˜π‘)
2019ghmplusg 19756 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
2114, 16, 18, 20syl3anc 1370 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
22 simpll 764 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 ∈ (𝑀 LMHom 𝑁))
23 simprl 768 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
24 simprr 770 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑦 ∈ (Baseβ€˜π‘€))
254, 6, 1, 2, 3lmhmlin 20791 . . . . . 6 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)(πΉβ€˜π‘¦)))
2622, 23, 24, 25syl3anc 1370 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)(πΉβ€˜π‘¦)))
27 simplr 766 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺 ∈ (𝑀 LMHom 𝑁))
284, 6, 1, 2, 3lmhmlin 20791 . . . . . 6 ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦)))
2927, 23, 24, 28syl3anc 1370 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦)))
3026, 29oveq12d 7430 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) + (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))) = ((π‘₯( ·𝑠 β€˜π‘)(πΉβ€˜π‘¦)) + (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦))))
319ad2antrr 723 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑁 ∈ LMod)
3211fveq2d 6896 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
3332ad2antrr 723 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
3423, 33eleqtrrd 2835 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)))
35 eqid 2731 . . . . . . . 8 (Baseβ€˜π‘) = (Baseβ€˜π‘)
361, 35lmhmf 20790 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3736ad2antrr 723 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3837, 24ffvelcdmd 7088 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘))
391, 35lmhmf 20790 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
4039ad2antlr 724 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
4140, 24ffvelcdmd 7088 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
42 eqid 2731 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘))
4335, 19, 5, 3, 42lmodvsdi 20640 . . . . 5 ((𝑁 ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)) ∧ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘) ∧ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))) β†’ (π‘₯( ·𝑠 β€˜π‘)((πΉβ€˜π‘¦) + (πΊβ€˜π‘¦))) = ((π‘₯( ·𝑠 β€˜π‘)(πΉβ€˜π‘¦)) + (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦))))
4431, 34, 38, 41, 43syl13anc 1371 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (π‘₯( ·𝑠 β€˜π‘)((πΉβ€˜π‘¦) + (πΊβ€˜π‘¦))) = ((π‘₯( ·𝑠 β€˜π‘)(πΉβ€˜π‘¦)) + (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦))))
4530, 44eqtr4d 2774 . . 3 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) + (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))) = (π‘₯( ·𝑠 β€˜π‘)((πΉβ€˜π‘¦) + (πΊβ€˜π‘¦))))
4637ffnd 6719 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 Fn (Baseβ€˜π‘€))
4740ffnd 6719 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺 Fn (Baseβ€˜π‘€))
48 fvexd 6907 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (Baseβ€˜π‘€) ∈ V)
497ad2antrr 723 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑀 ∈ LMod)
501, 4, 2, 6lmodvscl 20633 . . . . 5 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
5149, 23, 24, 50syl3anc 1370 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
52 fnfvof 7690 . . . 4 (((𝐹 Fn (Baseβ€˜π‘€) ∧ 𝐺 Fn (Baseβ€˜π‘€)) ∧ ((Baseβ€˜π‘€) ∈ V ∧ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘f + 𝐺)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = ((πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) + (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))))
5346, 47, 48, 51, 52syl22anc 836 . . 3 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘f + 𝐺)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = ((πΉβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) + (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))))
54 fnfvof 7690 . . . . 5 (((𝐹 Fn (Baseβ€˜π‘€) ∧ 𝐺 Fn (Baseβ€˜π‘€)) ∧ ((Baseβ€˜π‘€) ∈ V ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘f + 𝐺)β€˜π‘¦) = ((πΉβ€˜π‘¦) + (πΊβ€˜π‘¦)))
5546, 47, 48, 24, 54syl22anc 836 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘f + 𝐺)β€˜π‘¦) = ((πΉβ€˜π‘¦) + (πΊβ€˜π‘¦)))
5655oveq2d 7428 . . 3 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (π‘₯( ·𝑠 β€˜π‘)((𝐹 ∘f + 𝐺)β€˜π‘¦)) = (π‘₯( ·𝑠 β€˜π‘)((πΉβ€˜π‘¦) + (πΊβ€˜π‘¦))))
5745, 53, 563eqtr4d 2781 . 2 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘f + 𝐺)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)((𝐹 ∘f + 𝐺)β€˜π‘¦)))
581, 2, 3, 4, 5, 6, 8, 10, 12, 21, 57islmhmd 20795 1 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ∘f cof 7671  Basecbs 17149  +gcplusg 17202  Scalarcsca 17205   ·𝑠 cvsca 17206   GrpHom cghm 19128  Abelcabl 19691  LModclmod 20615   LMHom clmhm 20775
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-ghm 19129  df-cmn 19692  df-abl 19693  df-mgp 20030  df-ur 20077  df-ring 20130  df-lmod 20617  df-lmhm 20778
This theorem is referenced by:  nmhmplusg  24495  mendring  42237
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