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Theorem lmhmvsca 20307
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 2738 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 lmhmvsca.s . 2 · = ( ·𝑠𝑁)
4 eqid 2738 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 lmhmvsca.j . 2 𝐽 = (Scalar‘𝑁)
6 eqid 2738 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 20295 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
873ad2ant3 1134 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 20294 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
1093ad2ant3 1134 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 20292 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12113ad2ant3 1134 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
131fvexi 6788 . . . . . 6 𝑉 ∈ V
1413a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
15 simpl2 1191 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → 𝐴𝐾)
16 eqid 2738 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
171, 16lmhmf 20296 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
18173ad2ant3 1134 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
1918ffvelrnda 6961 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → (𝐹𝑣) ∈ (Base‘𝑁))
20 fconstmpt 5649 . . . . . 6 (𝑉 × {𝐴}) = (𝑣𝑉𝐴)
2120a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑉 × {𝐴}) = (𝑣𝑉𝐴))
2218feqmptd 6837 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣𝑉 ↦ (𝐹𝑣)))
2314, 15, 19, 21, 22offval2 7553 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
24 eqidd 2739 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
25 oveq2 7283 . . . . 5 (𝑢 = (𝐹𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹𝑣)))
2619, 22, 24, 25fmptco 7001 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
2723, 26eqtr4d 2781 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
28 simp2 1136 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴𝐾)
29 lmhmvsca.k . . . . . 6 𝐾 = (Base‘𝐽)
3016, 5, 3, 29lmodvsghm 20184 . . . . 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 20293 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
33323ad2ant3 1134 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34 ghmco 18854 . . . 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 2839 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 GrpHom 𝑁))
37 simpl1 1190 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐽 ∈ CRing)
38 simpl2 1191 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐴𝐾)
39 simprl 768 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
4012fveq2d 6778 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
4129, 40eqtrid 2790 . . . . . . . 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 2842 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥𝐾)
44 eqid 2738 . . . . . . 7 (.r𝐽) = (.r𝐽)
4529, 44crngcom 19801 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝑥𝐾) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4637, 38, 43, 45syl3anc 1370 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4746oveq1d 7290 . . . 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 770 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑦𝑉)
5149, 50ffvelrnd 6962 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹𝑦) ∈ (Base‘𝑁))
5216, 5, 3, 29, 44lmodvsass 20148 . . . . 5 ((𝑁 ∈ LMod ∧ (𝐴𝐾𝑥𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5348, 38, 43, 51, 52syl13anc 1371 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5416, 5, 3, 29, 44lmodvsass 20148 . . . . 5 ((𝑁 ∈ LMod ∧ (𝑥𝐾𝐴𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5548, 43, 38, 51, 54syl13anc 1371 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5647, 53, 553eqtr3d 2786 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴 · (𝑥 · (𝐹𝑦))) = (𝑥 · (𝐴 · (𝐹𝑦))))
571, 4, 2, 6lmodvscl 20140 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
58573expb 1119 . . . . 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 6601 . . . . . 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 20297 . . . . . . . 8 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
64633expb 1119 . . . . . . 7 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
65643ad2antl3 1186 . . . . . 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 7559 . . . 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 684 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
69 eqidd 2739 . . . . . 6 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (𝐹𝑦) = (𝐹𝑦))
7060, 38, 62, 69ofc1 7559 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7150, 70mpdan 684 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7271oveq2d 7291 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥 · (((𝑉 × {𝐴}) ∘f · 𝐹)‘𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
7356, 68, 723eqtr4d 2788 . 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 20301 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 1086   = wceq 1539  wcel 2106  Vcvv 3432  {csn 4561  cmpt 5157   × cxp 5587  ccom 5593   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  Basecbs 16912  .rcmulr 16963  Scalarcsca 16965   ·𝑠 cvsca 16966   GrpHom cghm 18831  CRingccrg 19784  LModclmod 20123   LMHom clmhm 20281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-ghm 18832  df-cmn 19388  df-mgp 19721  df-cring 19786  df-lmod 20125  df-lmhm 20284
This theorem is referenced by:  mendlmod  41018
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