MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmplusg Structured version   Visualization version   GIF version

Theorem lmhmplusg 20654
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 2732 . 2 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2732 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 eqid 2732 . 2 ( ·𝑠 β€˜π‘) = ( ·𝑠 β€˜π‘)
4 eqid 2732 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 eqid 2732 . 2 (Scalarβ€˜π‘) = (Scalarβ€˜π‘)
6 eqid 2732 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20643 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
87adantr 481 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20642 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝑁 ∈ LMod)
109adantr 481 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ LMod)
114, 5lmhmsca 20640 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
1211adantr 481 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
13 lmodabl 20518 . . . 4 (𝑁 ∈ LMod β†’ 𝑁 ∈ Abel)
1410, 13syl 17 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑁 ∈ Abel)
15 lmghm 20641 . . . 4 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
1615adantr 481 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐹 ∈ (𝑀 GrpHom 𝑁))
17 lmghm 20641 . . . 4 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
1817adantl 482 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
19 lmhmplusg.p . . . 4 + = (+gβ€˜π‘)
2019ghmplusg 19713 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
2114, 16, 18, 20syl3anc 1371 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
22 simpll 765 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 ∈ (𝑀 LMHom 𝑁))
23 simprl 769 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
24 simprr 771 . . . . . 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 20645 . . . . . 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 1371 . . . . 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 767 . . . . . 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 20645 . . . . . 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 1371 . . . . 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 7426 . . . 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 724 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑁 ∈ LMod)
3211fveq2d 6895 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
3332ad2antrr 724 . . . . . 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 2836 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)))
35 eqid 2732 . . . . . . . 8 (Baseβ€˜π‘) = (Baseβ€˜π‘)
361, 35lmhmf 20644 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) β†’ 𝐹:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3736ad2antrr 724 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3837, 24ffvelcdmd 7087 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π‘))
391, 35lmhmf 20644 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
4039ad2antlr 725 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
4140, 24ffvelcdmd 7087 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
42 eqid 2732 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘))
4335, 19, 5, 3, 42lmodvsdi 20494 . . . . 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 1372 . . . 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 2775 . . 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 6718 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 Fn (Baseβ€˜π‘€))
4740ffnd 6718 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺 Fn (Baseβ€˜π‘€))
48 fvexd 6906 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (Baseβ€˜π‘€) ∈ V)
497ad2antrr 724 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑀 ∈ LMod)
501, 4, 2, 6lmodvscl 20488 . . . . 5 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
5149, 23, 24, 50syl3anc 1371 . . . 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 7686 . . . 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 837 . . 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 7686 . . . . 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 837 . . . 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 7424 . . 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 2782 . 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 20649 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ∘f cof 7667  Basecbs 17143  +gcplusg 17196  Scalarcsca 17199   ·𝑠 cvsca 17200   GrpHom cghm 19088  Abelcabl 19648  LModclmod 20470   LMHom clmhm 20629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-ghm 19089  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-lmhm 20632
This theorem is referenced by:  nmhmplusg  24273  mendring  41924
  Copyright terms: Public domain W3C validator