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Theorem lmhmvsca 20506
Description: The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
lmhmvsca.v 𝑉 = (Base‘𝑀)
lmhmvsca.s · = ( ·𝑠𝑁)
lmhmvsca.j 𝐽 = (Scalar‘𝑁)
lmhmvsca.k 𝐾 = (Base‘𝐽)
Assertion
Ref Expression
lmhmvsca ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁))

Proof of Theorem lmhmvsca
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmvsca.v . 2 𝑉 = (Base‘𝑀)
2 eqid 2736 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 lmhmvsca.s . 2 · = ( ·𝑠𝑁)
4 eqid 2736 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 lmhmvsca.j . 2 𝐽 = (Scalar‘𝑁)
6 eqid 2736 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 20494 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
873ad2ant3 1135 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 20493 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
1093ad2ant3 1135 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 20491 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12113ad2ant3 1135 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
131fvexi 6856 . . . . . 6 𝑉 ∈ V
1413a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
15 simpl2 1192 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → 𝐴𝐾)
16 eqid 2736 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
171, 16lmhmf 20495 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
18173ad2ant3 1135 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
1918ffvelcdmda 7035 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → (𝐹𝑣) ∈ (Base‘𝑁))
20 fconstmpt 5694 . . . . . 6 (𝑉 × {𝐴}) = (𝑣𝑉𝐴)
2120a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑉 × {𝐴}) = (𝑣𝑉𝐴))
2218feqmptd 6910 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣𝑉 ↦ (𝐹𝑣)))
2314, 15, 19, 21, 22offval2 7637 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
24 eqidd 2737 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
25 oveq2 7365 . . . . 5 (𝑢 = (𝐹𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹𝑣)))
2619, 22, 24, 25fmptco 7075 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
2723, 26eqtr4d 2779 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
28 simp2 1137 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴𝐾)
29 lmhmvsca.k . . . . . 6 𝐾 = (Base‘𝐽)
3016, 5, 3, 29lmodvsghm 20383 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴𝐾) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
3110, 28, 30syl2anc 584 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
32 lmghm 20492 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
33323ad2ant3 1135 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34 ghmco 19028 . . . 4 (((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁) ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3531, 33, 34syl2anc 584 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3627, 35eqeltrd 2838 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 GrpHom 𝑁))
37 simpl1 1191 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐽 ∈ CRing)
38 simpl2 1192 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐴𝐾)
39 simprl 769 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
4012fveq2d 6846 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
4129, 40eqtrid 2788 . . . . . . . 8 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐾 = (Base‘(Scalar‘𝑀)))
4241adantr 481 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐾 = (Base‘(Scalar‘𝑀)))
4339, 42eleqtrrd 2841 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥𝐾)
44 eqid 2736 . . . . . . 7 (.r𝐽) = (.r𝐽)
4529, 44crngcom 19982 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝑥𝐾) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4637, 38, 43, 45syl3anc 1371 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4746oveq1d 7372 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)))
4810adantr 481 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑁 ∈ LMod)
4918adantr 481 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐹:𝑉⟶(Base‘𝑁))
50 simprr 771 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑦𝑉)
5149, 50ffvelcdmd 7036 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹𝑦) ∈ (Base‘𝑁))
5216, 5, 3, 29, 44lmodvsass 20347 . . . . 5 ((𝑁 ∈ LMod ∧ (𝐴𝐾𝑥𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5348, 38, 43, 51, 52syl13anc 1372 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5416, 5, 3, 29, 44lmodvsass 20347 . . . . 5 ((𝑁 ∈ LMod ∧ (𝑥𝐾𝐴𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5548, 43, 38, 51, 54syl13anc 1372 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5647, 53, 553eqtr3d 2784 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴 · (𝑥 · (𝐹𝑦))) = (𝑥 · (𝐴 · (𝐹𝑦))))
571, 4, 2, 6lmodvscl 20339 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
58573expb 1120 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
598, 58sylan 580 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
6013a1i 11 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑉 ∈ V)
6118ffnd 6669 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 Fn 𝑉)
6261adantr 481 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐹 Fn 𝑉)
634, 6, 1, 2, 3lmhmlin 20496 . . . . . . . 8 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
64633expb 1120 . . . . . . 7 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
65643ad2antl3 1187 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
6665adantr 481 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
6760, 38, 62, 66ofc1 7643 . . . 4 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
6859, 67mpdan 685 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
69 eqidd 2737 . . . . . 6 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (𝐹𝑦) = (𝐹𝑦))
7060, 38, 62, 69ofc1 7643 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7150, 70mpdan 685 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7271oveq2d 7373 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥 · (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
7356, 68, 723eqtr4d 2786 . 2 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦)))
741, 2, 3, 4, 5, 6, 8, 10, 12, 36, 73islmhmd 20500 1 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  {csn 4586  cmpt 5188   × cxp 5631  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  Basecbs 17083  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137   GrpHom cghm 19005  CRingccrg 19965  LModclmod 20322   LMHom clmhm 20480
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-cmn 19564  df-mgp 19897  df-cring 19967  df-lmod 20324  df-lmhm 20483
This theorem is referenced by:  mendlmod  41506
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