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Theorem dprdfeq0 19892
Description: The zero function is the only function that sums to zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0 0 = (0gβ€˜πΊ)
eldprdi.w π‘Š = {β„Ž ∈ X𝑖 ∈ 𝐼 (π‘†β€˜π‘–) ∣ β„Ž finSupp 0 }
eldprdi.1 (πœ‘ β†’ 𝐺dom DProd 𝑆)
eldprdi.2 (πœ‘ β†’ dom 𝑆 = 𝐼)
eldprdi.3 (πœ‘ β†’ 𝐹 ∈ π‘Š)
Assertion
Ref Expression
dprdfeq0 (πœ‘ β†’ ((𝐺 Ξ£g 𝐹) = 0 ↔ 𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 )))
Distinct variable groups:   π‘₯,β„Ž,𝐹   β„Ž,𝑖,𝐺,π‘₯   β„Ž,𝐼,𝑖,π‘₯   πœ‘,π‘₯   0 ,β„Ž,π‘₯   𝑆,β„Ž,𝑖,π‘₯
Allowed substitution hints:   πœ‘(β„Ž,𝑖)   𝐹(𝑖)   π‘Š(π‘₯,β„Ž,𝑖)   0 (𝑖)

Proof of Theorem dprdfeq0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldprdi.w . . . . . . 7 π‘Š = {β„Ž ∈ X𝑖 ∈ 𝐼 (π‘†β€˜π‘–) ∣ β„Ž finSupp 0 }
2 eldprdi.1 . . . . . . 7 (πœ‘ β†’ 𝐺dom DProd 𝑆)
3 eldprdi.2 . . . . . . 7 (πœ‘ β†’ dom 𝑆 = 𝐼)
4 eldprdi.3 . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ π‘Š)
5 eqid 2733 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
61, 2, 3, 4, 5dprdff 19882 . . . . . 6 (πœ‘ β†’ 𝐹:𝐼⟢(Baseβ€˜πΊ))
76feqmptd 6961 . . . . 5 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ (πΉβ€˜π‘₯)))
87adantr 482 . . . 4 ((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ (πΉβ€˜π‘₯)))
91, 2, 3, 4dprdfcl 19883 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ (π‘†β€˜π‘₯))
109adantlr 714 . . . . . . . 8 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ (π‘†β€˜π‘₯))
11 eldprdi.0 . . . . . . . . . . . 12 0 = (0gβ€˜πΊ)
122ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐺dom DProd 𝑆)
133ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ dom 𝑆 = 𝐼)
14 simpr 486 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ π‘₯ ∈ 𝐼)
15 eqid 2733 . . . . . . . . . . . . . 14 (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ))
1611, 1, 12, 13, 14, 10, 15dprdfid 19887 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∈ π‘Š ∧ (𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ))) = (πΉβ€˜π‘₯)))
1716simpld 496 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∈ π‘Š)
184ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐹 ∈ π‘Š)
19 eqid 2733 . . . . . . . . . . . 12 (-gβ€˜πΊ) = (-gβ€˜πΊ)
2011, 1, 12, 13, 17, 18, 19dprdfsub 19891 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∘f (-gβ€˜πΊ)𝐹) ∈ π‘Š ∧ (𝐺 Ξ£g ((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∘f (-gβ€˜πΊ)𝐹)) = ((𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )))(-gβ€˜πΊ)(𝐺 Ξ£g 𝐹))))
2120simprd 497 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g ((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∘f (-gβ€˜πΊ)𝐹)) = ((𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )))(-gβ€˜πΊ)(𝐺 Ξ£g 𝐹)))
222, 3dprddomcld 19871 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐼 ∈ V)
2322ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐼 ∈ V)
24 fvex 6905 . . . . . . . . . . . . . 14 (πΉβ€˜π‘₯) ∈ V
2511fvexi 6906 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4579 . . . . . . . . . . . . 13 if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ) ∈ V
2726a1i 11 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) β†’ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ) ∈ V)
28 fvexd 6907 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ V)
29 eqidd 2734 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )))
306ad2antrr 725 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐹:𝐼⟢(Baseβ€˜πΊ))
3130feqmptd 6961 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐹 = (𝑦 ∈ 𝐼 ↦ (πΉβ€˜π‘¦)))
3223, 27, 28, 29, 31offval2 7690 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∘f (-gβ€˜πΊ)𝐹) = (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦))))
3332oveq2d 7425 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g ((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∘f (-gβ€˜πΊ)𝐹)) = (𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)))))
3416simprd 497 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ))) = (πΉβ€˜π‘₯))
35 simplr 768 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g 𝐹) = 0 )
3634, 35oveq12d 7427 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )))(-gβ€˜πΊ)(𝐺 Ξ£g 𝐹)) = ((πΉβ€˜π‘₯)(-gβ€˜πΊ) 0 ))
37 dprdgrp 19875 . . . . . . . . . . . . 13 (𝐺dom DProd 𝑆 β†’ 𝐺 ∈ Grp)
3812, 37syl 17 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐺 ∈ Grp)
3930, 14ffvelcdmd 7088 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜πΊ))
405, 11, 19grpsubid1 18908 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜πΊ)) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ) 0 ) = (πΉβ€˜π‘₯))
4138, 39, 40syl2anc 585 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ) 0 ) = (πΉβ€˜π‘₯))
4236, 41eqtrd 2773 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )))(-gβ€˜πΊ)(𝐺 Ξ£g 𝐹)) = (πΉβ€˜π‘₯))
4321, 33, 423eqtr3d 2781 . . . . . . . . 9 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)))) = (πΉβ€˜π‘₯))
44 eqid 2733 . . . . . . . . . 10 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
45 grpmnd 18826 . . . . . . . . . . . 12 (𝐺 ∈ Grp β†’ 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ Mnd)
4746ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐺 ∈ Mnd)
485subgacs 19041 . . . . . . . . . . . . 13 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ (ACSβ€˜(Baseβ€˜πΊ)))
49 acsmre 17596 . . . . . . . . . . . . 13 ((SubGrpβ€˜πΊ) ∈ (ACSβ€˜(Baseβ€˜πΊ)) β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)))
5038, 48, 493syl 18 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)))
51 imassrn 6071 . . . . . . . . . . . . . 14 (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† ran 𝑆
522, 3dprdf2 19877 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5352ad2antrr 725 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5453frnd 6726 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ran 𝑆 βŠ† (SubGrpβ€˜πΊ))
55 mresspw 17536 . . . . . . . . . . . . . . . 16 ((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)) β†’ (SubGrpβ€˜πΊ) βŠ† 𝒫 (Baseβ€˜πΊ))
5650, 55syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (SubGrpβ€˜πΊ) βŠ† 𝒫 (Baseβ€˜πΊ))
5754, 56sstrd 3993 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ran 𝑆 βŠ† 𝒫 (Baseβ€˜πΊ))
5851, 57sstrid 3994 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† 𝒫 (Baseβ€˜πΊ))
59 sspwuni 5104 . . . . . . . . . . . . 13 ((𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† 𝒫 (Baseβ€˜πΊ) ↔ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† (Baseβ€˜πΊ))
6058, 59sylib 217 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† (Baseβ€˜πΊ))
61 eqid 2733 . . . . . . . . . . . . 13 (mrClsβ€˜(SubGrpβ€˜πΊ)) = (mrClsβ€˜(SubGrpβ€˜πΊ))
6261mrccl 17555 . . . . . . . . . . . 12 (((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)) ∧ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† (Baseβ€˜πΊ)) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubGrpβ€˜πΊ))
6350, 60, 62syl2anc 585 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubGrpβ€˜πΊ))
64 subgsubm 19028 . . . . . . . . . . 11 (((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubGrpβ€˜πΊ) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubMndβ€˜πΊ))
6563, 64syl 17 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubMndβ€˜πΊ))
66 oveq1 7416 . . . . . . . . . . . . 13 ((πΉβ€˜π‘₯) = if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘¦)) = (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)))
6766eleq1d 2819 . . . . . . . . . . . 12 ((πΉβ€˜π‘₯) = if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ) β†’ (((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ↔ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))))
68 oveq1 7416 . . . . . . . . . . . . 13 ( 0 = if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ) β†’ ( 0 (-gβ€˜πΊ)(πΉβ€˜π‘¦)) = (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)))
6968eleq1d 2819 . . . . . . . . . . . 12 ( 0 = if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 ) β†’ (( 0 (-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ↔ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))))
70 simpr 486 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ 𝑦 = π‘₯)
7170fveq2d 6896 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
7271oveq2d 7425 . . . . . . . . . . . . 13 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)))
735, 11, 19grpsubid 18907 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜πΊ)) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)) = 0 )
7438, 39, 73syl2anc 585 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)) = 0 )
7511subg0cl 19014 . . . . . . . . . . . . . . . 16 (((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubGrpβ€˜πΊ) β†’ 0 ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
7663, 75syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 0 ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
7774, 76eqeltrd 2834 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
7877ad2antrr 725 . . . . . . . . . . . . 13 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
7972, 78eqeltrd 2834 . . . . . . . . . . . 12 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
8063ad2antrr 725 . . . . . . . . . . . . 13 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubGrpβ€˜πΊ))
8180, 75syl 17 . . . . . . . . . . . . 13 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ 0 ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
8250, 61, 60mrcssidd 17569 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
8382ad2antrr 725 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
841, 12, 13, 18dprdfcl 19883 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
8584adantr 482 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
8653ffnd 6719 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝑆 Fn 𝐼)
8786ad2antrr 725 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ 𝑆 Fn 𝐼)
88 difssd 4133 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (𝐼 βˆ– {π‘₯}) βŠ† 𝐼)
89 df-ne 2942 . . . . . . . . . . . . . . . . . 18 (𝑦 β‰  π‘₯ ↔ Β¬ 𝑦 = π‘₯)
90 eldifsn 4791 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐼 βˆ– {π‘₯}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 β‰  π‘₯))
9190biimpri 227 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ 𝐼 ∧ 𝑦 β‰  π‘₯) β†’ 𝑦 ∈ (𝐼 βˆ– {π‘₯}))
9289, 91sylan2br 596 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ 𝐼 ∧ Β¬ 𝑦 = π‘₯) β†’ 𝑦 ∈ (𝐼 βˆ– {π‘₯}))
9392adantll 713 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ 𝑦 ∈ (𝐼 βˆ– {π‘₯}))
94 fnfvima 7235 . . . . . . . . . . . . . . . 16 ((𝑆 Fn 𝐼 ∧ (𝐼 βˆ– {π‘₯}) βŠ† 𝐼 ∧ 𝑦 ∈ (𝐼 βˆ– {π‘₯})) β†’ (π‘†β€˜π‘¦) ∈ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))
9587, 88, 93, 94syl3anc 1372 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (π‘†β€˜π‘¦) ∈ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))
96 elunii 4914 . . . . . . . . . . . . . . 15 (((πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦) ∧ (π‘†β€˜π‘¦) ∈ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) β†’ (πΉβ€˜π‘¦) ∈ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))
9785, 95, 96syl2anc 585 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (πΉβ€˜π‘¦) ∈ βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))
9883, 97sseldd 3984 . . . . . . . . . . . . 13 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (πΉβ€˜π‘¦) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
9919subgsubcl 19017 . . . . . . . . . . . . 13 ((((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∈ (SubGrpβ€˜πΊ) ∧ 0 ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))) ∧ (πΉβ€˜π‘¦) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))) β†’ ( 0 (-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
10080, 81, 98, 99syl3anc 1372 . . . . . . . . . . . 12 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ ( 0 (-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
10167, 69, 79, 100ifbothda 4567 . . . . . . . . . . 11 ((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) β†’ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
102101fmpttd 7115 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦))):𝐼⟢((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
10320simpld 496 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∘f (-gβ€˜πΊ)𝐹) ∈ π‘Š)
10432, 103eqeltrrd 2835 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦))) ∈ π‘Š)
1051, 12, 13, 104, 44dprdfcntz 19885 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ran (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦))) βŠ† ((Cntzβ€˜πΊ)β€˜ran (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)))))
1061, 12, 13, 104dprdffsupp 19884 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦))) finSupp 0 )
10711, 44, 47, 23, 65, 102, 105, 106gsumzsubmcl 19786 . . . . . . . . 9 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g (𝑦 ∈ 𝐼 ↦ (if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )(-gβ€˜πΊ)(πΉβ€˜π‘¦)))) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
10843, 107eqeltrrd 2835 . . . . . . . 8 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯}))))
10910, 108elind 4195 . . . . . . 7 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ ((π‘†β€˜π‘₯) ∩ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))))
11012, 13, 14, 11, 61dprddisj 19879 . . . . . . 7 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((π‘†β€˜π‘₯) ∩ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ (𝑆 β€œ (𝐼 βˆ– {π‘₯})))) = { 0 })
111109, 110eleqtrd 2836 . . . . . 6 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ { 0 })
112 elsni 4646 . . . . . 6 ((πΉβ€˜π‘₯) ∈ { 0 } β†’ (πΉβ€˜π‘₯) = 0 )
113111, 112syl 17 . . . . 5 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) = 0 )
114113mpteq2dva 5249 . . . 4 ((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) β†’ (π‘₯ ∈ 𝐼 ↦ (πΉβ€˜π‘₯)) = (π‘₯ ∈ 𝐼 ↦ 0 ))
1158, 114eqtrd 2773 . . 3 ((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 ))
116115ex 414 . 2 (πœ‘ β†’ ((𝐺 Ξ£g 𝐹) = 0 β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 )))
11711gsumz 18717 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )) = 0 )
11846, 22, 117syl2anc 585 . . 3 (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )) = 0 )
119 oveq2 7417 . . . 4 (𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 ) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )))
120119eqeq1d 2735 . . 3 (𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 ) β†’ ((𝐺 Ξ£g 𝐹) = 0 ↔ (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )) = 0 ))
121118, 120syl5ibrcom 246 . 2 (πœ‘ β†’ (𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 ) β†’ (𝐺 Ξ£g 𝐹) = 0 ))
122116, 121impbid 211 1 (πœ‘ β†’ ((𝐺 Ξ£g 𝐹) = 0 ↔ 𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 )))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  ifcif 4529  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668  Xcixp 8891   finSupp cfsupp 9361  Basecbs 17144  0gc0g 17385   Ξ£g cgsu 17386  Moorecmre 17526  mrClscmrc 17527  ACScacs 17529  Mndcmnd 18625  SubMndcsubmnd 18670  Grpcgrp 18819  -gcsg 18821  SubGrpcsubg 19000  Cntzccntz 19179   DProd cdprd 19863
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-gsum 17388  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-ghm 19090  df-gim 19133  df-cntz 19181  df-oppg 19210  df-cmn 19650  df-dprd 19865
This theorem is referenced by:  dprdf11  19893
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