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Theorem lmhmeql 20532
Description: The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypothesis
Ref Expression
lmhmeql.u π‘ˆ = (LSubSpβ€˜π‘†)
Assertion
Ref Expression
lmhmeql ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ π‘ˆ)

Proof of Theorem lmhmeql
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 20508 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 lmghm 20508 . . 3 (𝐺 ∈ (𝑆 LMHom 𝑇) β†’ 𝐺 ∈ (𝑆 GrpHom 𝑇))
3 ghmeql 19038 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†))
41, 2, 3syl2an 597 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†))
5 fveq2 6847 . . . . . . . 8 (𝑧 = (π‘₯( ·𝑠 β€˜π‘†)𝑦) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)))
6 fveq2 6847 . . . . . . . 8 (𝑧 = (π‘₯( ·𝑠 β€˜π‘†)𝑦) β†’ (πΊβ€˜π‘§) = (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)))
75, 6eqeq12d 2753 . . . . . . 7 (𝑧 = (π‘₯( ·𝑠 β€˜π‘†)𝑦) β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)) = (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦))))
8 lmhmlmod1 20510 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
98adantr 482 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ 𝑆 ∈ LMod)
109ad2antrr 725 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝑆 ∈ LMod)
11 simplr 768 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†)))
12 simprl 770 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝑦 ∈ (Baseβ€˜π‘†))
13 eqid 2737 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
14 eqid 2737 . . . . . . . . 9 (Scalarβ€˜π‘†) = (Scalarβ€˜π‘†)
15 eqid 2737 . . . . . . . . 9 ( ·𝑠 β€˜π‘†) = ( ·𝑠 β€˜π‘†)
16 eqid 2737 . . . . . . . . 9 (Baseβ€˜(Scalarβ€˜π‘†)) = (Baseβ€˜(Scalarβ€˜π‘†))
1713, 14, 15, 16lmodvscl 20355 . . . . . . . 8 ((𝑆 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
1810, 11, 12, 17syl3anc 1372 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
19 oveq2 7370 . . . . . . . . 9 ((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘¦)) = (π‘₯( ·𝑠 β€˜π‘‡)(πΊβ€˜π‘¦)))
2019ad2antll 728 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (π‘₯( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘¦)) = (π‘₯( ·𝑠 β€˜π‘‡)(πΊβ€˜π‘¦)))
21 simplll 774 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))
22 eqid 2737 . . . . . . . . . 10 ( ·𝑠 β€˜π‘‡) = ( ·𝑠 β€˜π‘‡)
2314, 16, 13, 15, 22lmhmlin 20512 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)) = (π‘₯( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘¦)))
2421, 11, 12, 23syl3anc 1372 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)) = (π‘₯( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘¦)))
25 simpllr 775 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝐺 ∈ (𝑆 LMHom 𝑇))
2614, 16, 13, 15, 22lmhmlin 20512 . . . . . . . . 9 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)) = (π‘₯( ·𝑠 β€˜π‘‡)(πΊβ€˜π‘¦)))
2725, 11, 12, 26syl3anc 1372 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)) = (π‘₯( ·𝑠 β€˜π‘‡)(πΊβ€˜π‘¦)))
2820, 24, 273eqtr4d 2787 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)) = (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘†)𝑦)))
297, 18, 28elrabd 3652 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
3029expr 458 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ ((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
3130ralrimiva 3144 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) β†’ βˆ€π‘¦ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
32 eqid 2737 . . . . . . . . 9 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
3313, 32lmhmf 20511 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3433ffnd 6674 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 Fn (Baseβ€˜π‘†))
3513, 32lmhmf 20511 . . . . . . . 8 (𝐺 ∈ (𝑆 LMHom 𝑇) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
3635ffnd 6674 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) β†’ 𝐺 Fn (Baseβ€˜π‘†))
37 fndmin 7000 . . . . . . 7 ((𝐹 Fn (Baseβ€˜π‘†) ∧ 𝐺 Fn (Baseβ€˜π‘†)) β†’ dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
3834, 36, 37syl2an 597 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
3938adantr 482 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) β†’ dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
40 eleq2 2827 . . . . . . 7 (dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} β†’ ((π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
4140raleqbi1dv 3310 . . . . . 6 (dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} β†’ (βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
42 fveq2 6847 . . . . . . . 8 (𝑧 = 𝑦 β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
43 fveq2 6847 . . . . . . . 8 (𝑧 = 𝑦 β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘¦))
4442, 43eqeq12d 2753 . . . . . . 7 (𝑧 = 𝑦 β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)))
4544ralrab 3656 . . . . . 6 (βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
4641, 45bitrdi 287 . . . . 5 (dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} β†’ (βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})))
4739, 46syl 17 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) β†’ (βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})))
4831, 47mpbird 257 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))) β†’ βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺))
4948ralrimiva 3144 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺))
50 lmhmeql.u . . . 4 π‘ˆ = (LSubSpβ€˜π‘†)
5114, 16, 13, 15, 50islss4 20439 . . 3 (𝑆 ∈ LMod β†’ (dom (𝐹 ∩ 𝐺) ∈ π‘ˆ ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺))))
529, 51syl 17 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ (dom (𝐹 ∩ 𝐺) ∈ π‘ˆ ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubGrpβ€˜π‘†) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘†))βˆ€π‘¦ ∈ dom (𝐹 ∩ 𝐺)(π‘₯( ·𝑠 β€˜π‘†)𝑦) ∈ dom (𝐹 ∩ 𝐺))))
534, 49, 52mpbir2and 712 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410   ∩ cin 3914  dom cdm 5638   Fn wfn 6496  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  Scalarcsca 17143   ·𝑠 cvsca 17144  SubGrpcsubg 18929   GrpHom cghm 19012  LModclmod 20338  LSubSpclss 20408   LMHom clmhm 20496
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-ghm 19013  df-mgp 19904  df-ur 19921  df-ring 19973  df-lmod 20340  df-lss 20409  df-lmhm 20499
This theorem is referenced by:  lspextmo  20533
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