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Theorem lmhmvsca 18959
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 2626 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 lmhmvsca.s . 2 · = ( ·𝑠𝑁)
4 eqid 2626 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 lmhmvsca.j . 2 𝐽 = (Scalar‘𝑁)
6 eqid 2626 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 18947 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
873ad2ant3 1082 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 18946 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
1093ad2ant3 1082 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 18944 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐽 = (Scalar‘𝑀))
12113ad2ant3 1082 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐽 = (Scalar‘𝑀))
13 fvex 6160 . . . . . . 7 (Base‘𝑀) ∈ V
141, 13eqeltri 2700 . . . . . 6 𝑉 ∈ V
1514a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝑉 ∈ V)
16 simpl2 1063 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → 𝐴𝐾)
17 eqid 2626 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
181, 17lmhmf 18948 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:𝑉⟶(Base‘𝑁))
19183ad2ant3 1082 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹:𝑉⟶(Base‘𝑁))
2019ffvelrnda 6316 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑣𝑉) → (𝐹𝑣) ∈ (Base‘𝑁))
21 fconstmpt 5128 . . . . . 6 (𝑉 × {𝐴}) = (𝑣𝑉𝐴)
2221a1i 11 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑉 × {𝐴}) = (𝑣𝑉𝐴))
2319feqmptd 6207 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 = (𝑣𝑉 ↦ (𝐹𝑣)))
2415, 16, 20, 22, 23offval2 6868 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
25 eqidd 2627 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) = (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)))
26 oveq2 6613 . . . . 5 (𝑢 = (𝐹𝑣) → (𝐴 · 𝑢) = (𝐴 · (𝐹𝑣)))
2720, 23, 25, 26fmptco 6352 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) = (𝑣𝑉 ↦ (𝐴 · (𝐹𝑣))))
2824, 27eqtr4d 2663 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) = ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹))
29 simp2 1060 . . . . 5 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐴𝐾)
30 lmhmvsca.k . . . . . 6 𝐾 = (Base‘𝐽)
3117, 5, 3, 30lmodvsghm 18840 . . . . 5 ((𝑁 ∈ LMod ∧ 𝐴𝐾) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
3210, 29, 31syl2anc 692 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∈ (𝑁 GrpHom 𝑁))
33 lmghm 18945 . . . . 5 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
34333ad2ant3 1082 . . . 4 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
35 ghmco 17596 . . . 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 692 . . 3 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑢 ∈ (Base‘𝑁) ↦ (𝐴 · 𝑢)) ∘ 𝐹) ∈ (𝑀 GrpHom 𝑁))
3728, 36eqeltrd 2704 . 2 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) ∈ (𝑀 GrpHom 𝑁))
38 simpl1 1062 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐽 ∈ CRing)
39 simpl2 1063 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐴𝐾)
40 simprl 793 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
4112fveq2d 6154 . . . . . . . . 9 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → (Base‘𝐽) = (Base‘(Scalar‘𝑀)))
4230, 41syl5eq 2672 . . . . . . . 8 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐾 = (Base‘(Scalar‘𝑀)))
4342adantr 481 . . . . . . 7 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐾 = (Base‘(Scalar‘𝑀)))
4440, 43eleqtrrd 2707 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑥𝐾)
45 eqid 2626 . . . . . . 7 (.r𝐽) = (.r𝐽)
4630, 45crngcom 18478 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝑥𝐾) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4738, 39, 44, 46syl3anc 1323 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴(.r𝐽)𝑥) = (𝑥(.r𝐽)𝐴))
4847oveq1d 6620 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)))
4910adantr 481 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑁 ∈ LMod)
5019adantr 481 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐹:𝑉⟶(Base‘𝑁))
51 simprr 795 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝑦𝑉)
5250, 51ffvelrnd 6317 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹𝑦) ∈ (Base‘𝑁))
5317, 5, 3, 30, 45lmodvsass 18804 . . . . 5 ((𝑁 ∈ LMod ∧ (𝐴𝐾𝑥𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5449, 39, 44, 52, 53syl13anc 1325 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝐴(.r𝐽)𝑥) · (𝐹𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
5517, 5, 3, 30, 45lmodvsass 18804 . . . . 5 ((𝑁 ∈ LMod ∧ (𝑥𝐾𝐴𝐾 ∧ (𝐹𝑦) ∈ (Base‘𝑁))) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5649, 44, 39, 52, 55syl13anc 1325 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → ((𝑥(.r𝐽)𝐴) · (𝐹𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
5748, 54, 563eqtr3d 2668 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐴 · (𝑥 · (𝐹𝑦))) = (𝑥 · (𝐴 · (𝐹𝑦))))
581, 4, 2, 6lmodvscl 18796 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
59583expb 1263 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥( ·𝑠𝑀)𝑦) ∈ 𝑉)
608, 59sylan 488 . . . 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 6004 . . . . . . 7 (𝐹:𝑉⟶(Base‘𝑁) → 𝐹 Fn 𝑉)
6319, 62syl 17 . . . . . 6 ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → 𝐹 Fn 𝑉)
6463adantr 481 . . . . 5 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → 𝐹 Fn 𝑉)
654, 6, 1, 2, 3lmhmlin 18949 . . . . . . . 8 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
66653expb 1263 . . . . . . 7 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
67663ad2antl3 1223 . . . . . 6 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥 · (𝐹𝑦)))
6867adantr 481 . . . . 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 6874 . . . 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 701 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐴 · (𝑥 · (𝐹𝑦))))
71 eqidd 2627 . . . . . 6 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (𝐹𝑦) = (𝐹𝑦))
7261, 39, 64, 71ofc1 6874 . . . . 5 ((((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) ∧ 𝑦𝑉) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7351, 72mpdan 701 . . . 4 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦) = (𝐴 · (𝐹𝑦)))
7473oveq2d 6621 . . 3 (((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦𝑉)) → (𝑥 · (((𝑉 × {𝐴}) ∘𝑓 · 𝐹)‘𝑦)) = (𝑥 · (𝐴 · (𝐹𝑦))))
7557, 70, 743eqtr4d 2670 . 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 18953 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 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cmpt 4678   × cxp 5077  ccom 5083   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  𝑓 cof 6849  Basecbs 15776  .rcmulr 15858  Scalarcsca 15860   ·𝑠 cvsca 15861   GrpHom cghm 17573  CRingccrg 18464  LModclmod 18779   LMHom clmhm 18933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-grp 17341  df-ghm 17574  df-cmn 18111  df-mgp 18406  df-cring 18466  df-lmod 18781  df-lmhm 18936
This theorem is referenced by:  mendlmod  37230
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