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Theorem lmhmvsca 18866
 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 2610 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 lmhmvsca.s . 2 · = ( ·𝑠𝑁)
4 eqid 2610 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 lmhmvsca.j . 2 𝐽 = (Scalar‘𝑁)
6 eqid 2610 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 18854 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
873ad2ant3 1077 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 18853 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
1093ad2ant3 1077 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 18851 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12113ad2ant3 1077 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
13 fvex 6113 . . . . . . 7 (Base‘𝑀) ∈ V
141, 13eqeltri 2684 . . . . . 6 𝑉 ∈ V
1514a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
16 simpl2 1058 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → 𝐴𝐾)
17 eqid 2610 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
181, 17lmhmf 18855 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
19183ad2ant3 1077 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
2019ffvelrnda 6267 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → (𝐹𝑣) ∈ (Base‘𝑁))
21 fconstmpt 5085 . . . . . 6 (𝑉 × {𝐴}) = (𝑣𝑉𝐴)
2221a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑉 × {𝐴}) = (𝑣𝑉𝐴))
2319feqmptd 6159 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣𝑉 ↦ (𝐹𝑣)))
2415, 16, 20, 22, 23offval2 6812 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
25 eqidd 2611 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
26 oveq2 6557 . . . . 5 (𝑢 = (𝐹𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹𝑣)))
2720, 23, 25, 26fmptco 6303 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
2824, 27eqtr4d 2647 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
29 simp2 1055 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴𝐾)
30 lmhmvsca.k . . . . . 6 𝐾 = (Base‘𝐽)
3117, 5, 3, 30lmodvsghm 18747 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴𝐾) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
3210, 29, 31syl2anc 691 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
33 lmghm 18852 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34333ad2ant3 1077 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
35 ghmco 17503 . . . 4 (((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁) ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3632, 34, 35syl2anc 691 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3728, 36eqeltrd 2688 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) ∈ (𝑀 GrpHom 𝑁))
38 simpl1 1057 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐽 ∈ CRing)
39 simpl2 1058 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐴𝐾)
40 simprl 790 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
4112fveq2d 6107 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
4230, 41syl5eq 2656 . . . . . . . 8 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐾 = (Base‘(Scalar‘𝑀)))
4342adantr 480 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐾 = (Base‘(Scalar‘𝑀)))
4440, 43eleqtrrd 2691 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥𝐾)
45 eqid 2610 . . . . . . 7 (.r𝐽) = (.r𝐽)
4630, 45crngcom 18385 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝑥𝐾) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4738, 39, 44, 46syl3anc 1318 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4847oveq1d 6564 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)))
4910adantr 480 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑁 ∈ LMod)
5019adantr 480 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐹:𝑉⟶(Base‘𝑁))
51 simprr 792 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑦𝑉)
5250, 51ffvelrnd 6268 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹𝑦) ∈ (Base‘𝑁))
5317, 5, 3, 30, 45lmodvsass 18711 . . . . 5 ((𝑁 ∈ LMod ∧ (𝐴𝐾𝑥𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5449, 39, 44, 52, 53syl13anc 1320 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5517, 5, 3, 30, 45lmodvsass 18711 . . . . 5 ((𝑁 ∈ LMod ∧ (𝑥𝐾𝐴𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5649, 44, 39, 52, 55syl13anc 1320 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5748, 54, 563eqtr3d 2652 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴 · (𝑥 · (𝐹𝑦))) = (𝑥 · (𝐴 · (𝐹𝑦))))
581, 4, 2, 6lmodvscl 18703 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
59583expb 1258 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
608, 59sylan 487 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
6114a1i 11 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑉 ∈ V)
62 ffn 5958 . . . . . . 7 (𝐹:𝑉⟶(Base‘𝑁) → 𝐹 Fn 𝑉)
6319, 62syl 17 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 Fn 𝑉)
6463adantr 480 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐹 Fn 𝑉)
654, 6, 1, 2, 3lmhmlin 18856 . . . . . . . 8 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
66653expb 1258 . . . . . . 7 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
67663ad2antl3 1218 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
6867adantr 480 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
6961, 39, 64, 68ofc1 6818 . . . 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))))
7060, 69mpdan 699 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
71 eqidd 2611 . . . . . 6 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (𝐹𝑦) = (𝐹𝑦))
7261, 39, 64, 71ofc1 6818 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7351, 72mpdan 699 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7473oveq2d 6565 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥 · (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
7557, 70, 743eqtr4d 2654 . 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)))
761, 2, 3, 4, 5, 6, 8, 10, 12, 37, 75islmhmd 18860 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772   GrpHom cghm 17480  CRingccrg 18371  LModclmod 18686   LMHom clmhm 18840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-cmn 18018  df-mgp 18313  df-cring 18373  df-lmod 18688  df-lmhm 18843 This theorem is referenced by:  mendlmod  36782
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