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Theorem lmhmplusg 20799
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 2730 . 2 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2730 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 eqid 2730 . 2 ( ·𝑠 β€˜π‘) = ( ·𝑠 β€˜π‘)
4 eqid 2730 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 eqid 2730 . 2 (Scalarβ€˜π‘) = (Scalarβ€˜π‘)
6 eqid 2730 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20788 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
87adantr 479 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20787 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
109adantr 479 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20785 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
1211adantr 479 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
13 lmodabl 20663 . . . 4 (𝑁 ∈ LMod β†’ 𝑁 ∈ Abel)
1410, 13syl 17 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ Abel)
15 lmghm 20786 . . . 4 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
1615adantr 479 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
17 lmghm 20786 . . . 4 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
1817adantl 480 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
19 lmhmplusg.p . . . 4 + = (+gβ€˜π‘)
2019ghmplusg 19755 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
2114, 16, 18, 20syl3anc 1369 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
22 simpll 763 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 ∈ (𝑀 LMHom 𝑁))
23 simprl 767 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
24 simprr 769 . . . . . 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 20790 . . . . . 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 1369 . . . . 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 765 . . . . . 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 20790 . . . . . 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 1369 . . . . 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 7429 . . . 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 722 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑁 ∈ LMod)
3211fveq2d 6894 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
3332ad2antrr 722 . . . . . 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 2834 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)))
35 eqid 2730 . . . . . . . 8 (Baseβ€˜π‘) = (Baseβ€˜π‘)
361, 35lmhmf 20789 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3736ad2antrr 722 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3837, 24ffvelcdmd 7086 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘))
391, 35lmhmf 20789 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
4039ad2antlr 723 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
4140, 24ffvelcdmd 7086 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
42 eqid 2730 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘))
4335, 19, 5, 3, 42lmodvsdi 20639 . . . . 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 1370 . . . 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 2773 . . 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 6717 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 Fn (Baseβ€˜π‘€))
4740ffnd 6717 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺 Fn (Baseβ€˜π‘€))
48 fvexd 6905 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (Baseβ€˜π‘€) ∈ V)
497ad2antrr 722 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑀 ∈ LMod)
501, 4, 2, 6lmodvscl 20632 . . . . 5 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
5149, 23, 24, 50syl3anc 1369 . . . 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 7689 . . . 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 835 . . 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 7689 . . . . 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 835 . . . 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 7427 . . 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 2780 . 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 20794 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∘f cof 7670  Basecbs 17148  +gcplusg 17201  Scalarcsca 17204   ·𝑠 cvsca 17205   GrpHom cghm 19127  Abelcabl 19690  LModclmod 20614   LMHom clmhm 20774
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-ghm 19128  df-cmn 19691  df-abl 19692  df-mgp 20029  df-ur 20076  df-ring 20129  df-lmod 20616  df-lmhm 20777
This theorem is referenced by:  nmhmplusg  24494  mendring  42236
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