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Theorem lmhmvsca 19093
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 2651 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 lmhmvsca.s . 2 · = ( ·𝑠𝑁)
4 eqid 2651 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 lmhmvsca.j . 2 𝐽 = (Scalar‘𝑁)
6 eqid 2651 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 19081 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
873ad2ant3 1104 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 19080 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
1093ad2ant3 1104 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 19078 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12113ad2ant3 1104 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
13 fvex 6239 . . . . . . 7 (Base‘𝑀) ∈ V
141, 13eqeltri 2726 . . . . . 6 𝑉 ∈ V
1514a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
16 simpl2 1085 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → 𝐴𝐾)
17 eqid 2651 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
181, 17lmhmf 19082 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
19183ad2ant3 1104 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
2019ffvelrnda 6399 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → (𝐹𝑣) ∈ (Base‘𝑁))
21 fconstmpt 5197 . . . . . 6 (𝑉 × {𝐴}) = (𝑣𝑉𝐴)
2221a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑉 × {𝐴}) = (𝑣𝑉𝐴))
2319feqmptd 6288 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣𝑉 ↦ (𝐹𝑣)))
2415, 16, 20, 22, 23offval2 6956 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
25 eqidd 2652 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
26 oveq2 6698 . . . . 5 (𝑢 = (𝐹𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹𝑣)))
2720, 23, 25, 26fmptco 6436 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
2824, 27eqtr4d 2688 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
29 simp2 1082 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴𝐾)
30 lmhmvsca.k . . . . . 6 𝐾 = (Base‘𝐽)
3117, 5, 3, 30lmodvsghm 18972 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴𝐾) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
3210, 29, 31syl2anc 694 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
33 lmghm 19079 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34333ad2ant3 1104 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
35 ghmco 17727 . . . 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 694 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3728, 36eqeltrd 2730 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) ∈ (𝑀 GrpHom 𝑁))
38 simpl1 1084 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐽 ∈ CRing)
39 simpl2 1085 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐴𝐾)
40 simprl 809 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
4112fveq2d 6233 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
4230, 41syl5eq 2697 . . . . . . . 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 2733 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥𝐾)
45 eqid 2651 . . . . . . 7 (.r𝐽) = (.r𝐽)
4630, 45crngcom 18608 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝑥𝐾) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4738, 39, 44, 46syl3anc 1366 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4847oveq1d 6705 . . . 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 811 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑦𝑉)
5250, 51ffvelrnd 6400 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹𝑦) ∈ (Base‘𝑁))
5317, 5, 3, 30, 45lmodvsass 18936 . . . . 5 ((𝑁 ∈ LMod ∧ (𝐴𝐾𝑥𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5449, 39, 44, 52, 53syl13anc 1368 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5517, 5, 3, 30, 45lmodvsass 18936 . . . . 5 ((𝑁 ∈ LMod ∧ (𝑥𝐾𝐴𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5649, 44, 39, 52, 55syl13anc 1368 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5748, 54, 563eqtr3d 2693 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴 · (𝑥 · (𝐹𝑦))) = (𝑥 · (𝐴 · (𝐹𝑦))))
581, 4, 2, 6lmodvscl 18928 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
59583expb 1285 . . . . 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 6083 . . . . . . 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 19083 . . . . . . . 8 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
66653expb 1285 . . . . . . 7 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
67663ad2antl3 1245 . . . . . 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 6962 . . . 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 703 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
71 eqidd 2652 . . . . . 6 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (𝐹𝑦) = (𝐹𝑦))
7261, 39, 64, 71ofc1 6962 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7351, 72mpdan 703 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7473oveq2d 6706 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥 · (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
7557, 70, 743eqtr4d 2695 . 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 19087 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 1054   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cmpt 4762   × cxp 5141  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  Basecbs 15904  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992   GrpHom cghm 17704  CRingccrg 18594  LModclmod 18911   LMHom clmhm 19067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-ghm 17705  df-cmn 18241  df-mgp 18536  df-cring 18596  df-lmod 18913  df-lmhm 19070
This theorem is referenced by:  mendlmod  38080
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