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Theorem dprdfeq0 19894
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 2732 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
61, 2, 3, 4, 5dprdff 19884 . . . . . 6 (πœ‘ β†’ 𝐹:𝐼⟢(Baseβ€˜πΊ))
76feqmptd 6960 . . . . 5 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ (πΉβ€˜π‘₯)))
87adantr 481 . . . 4 ((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ (πΉβ€˜π‘₯)))
91, 2, 3, 4dprdfcl 19885 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ (π‘†β€˜π‘₯))
109adantlr 713 . . . . . . . 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 724 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐺dom DProd 𝑆)
133ad2antrr 724 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ dom 𝑆 = 𝐼)
14 simpr 485 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ π‘₯ ∈ 𝐼)
15 eqid 2732 . . . . . . . . . . . . . 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 19889 . . . . . . . . . . . . 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 495 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑦 ∈ 𝐼 ↦ if(𝑦 = π‘₯, (πΉβ€˜π‘₯), 0 )) ∈ π‘Š)
184ad2antrr 724 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐹 ∈ π‘Š)
19 eqid 2732 . . . . . . . . . . . 12 (-gβ€˜πΊ) = (-gβ€˜πΊ)
2011, 1, 12, 13, 17, 18, 19dprdfsub 19893 . . . . . . . . . . 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 496 . . . . . . . . . 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 19873 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐼 ∈ V)
2322ad2antrr 724 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐼 ∈ V)
24 fvex 6904 . . . . . . . . . . . . . 14 (πΉβ€˜π‘₯) ∈ V
2511fvexi 6905 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4578 . . . . . . . . . . . . 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 6906 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ V)
29 eqidd 2733 . . . . . . . . . . . 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 724 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐹:𝐼⟢(Baseβ€˜πΊ))
3130feqmptd 6960 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐹 = (𝑦 ∈ 𝐼 ↦ (πΉβ€˜π‘¦)))
3223, 27, 28, 29, 31offval2 7692 . . . . . . . . . . 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 7427 . . . . . . . . . 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 496 . . . . . . . . . . . 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 767 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝐺 Ξ£g 𝐹) = 0 )
3634, 35oveq12d 7429 . . . . . . . . . . 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 19877 . . . . . . . . . . . . 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 7087 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜πΊ))
405, 11, 19grpsubid1 18910 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜πΊ)) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ) 0 ) = (πΉβ€˜π‘₯))
4138, 39, 40syl2anc 584 . . . . . . . . . . 11 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ) 0 ) = (πΉβ€˜π‘₯))
4236, 41eqtrd 2772 . . . . . . . . . 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 2780 . . . . . . . . 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 2732 . . . . . . . . . 10 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
45 grpmnd 18828 . . . . . . . . . . . 12 (𝐺 ∈ Grp β†’ 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ Mnd)
4746ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝐺 ∈ Mnd)
485subgacs 19043 . . . . . . . . . . . . 13 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ (ACSβ€˜(Baseβ€˜πΊ)))
49 acsmre 17598 . . . . . . . . . . . . 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 6070 . . . . . . . . . . . . . 14 (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† ran 𝑆
522, 3dprdf2 19879 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5352ad2antrr 724 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝑆:𝐼⟢(SubGrpβ€˜πΊ))
5453frnd 6725 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ran 𝑆 βŠ† (SubGrpβ€˜πΊ))
55 mresspw 17538 . . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ran 𝑆 βŠ† 𝒫 (Baseβ€˜πΊ))
5851, 57sstrid 3993 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (𝑆 β€œ (𝐼 βˆ– {π‘₯})) βŠ† 𝒫 (Baseβ€˜πΊ))
59 sspwuni 5103 . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . 13 (mrClsβ€˜(SubGrpβ€˜πΊ)) = (mrClsβ€˜(SubGrpβ€˜πΊ))
6261mrccl 17557 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 19030 . . . . . . . . . . 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 7418 . . . . . . . . . . . . 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 2818 . . . . . . . . . . . 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 7418 . . . . . . . . . . . . 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 2818 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ 𝑦 = π‘₯)
7170fveq2d 6895 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = π‘₯) β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯))
7271oveq2d 7427 . . . . . . . . . . . . 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 18909 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (πΉβ€˜π‘₯) ∈ (Baseβ€˜πΊ)) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)) = 0 )
7438, 39, 73syl2anc 584 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ ((πΉβ€˜π‘₯)(-gβ€˜πΊ)(πΉβ€˜π‘₯)) = 0 )
7511subg0cl 19016 . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . 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 2833 . . . . . . . . . . . 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 724 . . . . . . . . . . . . 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 17571 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 19885 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
8584adantr 481 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (πΉβ€˜π‘¦) ∈ (π‘†β€˜π‘¦))
8653ffnd 6718 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ 𝑆 Fn 𝐼)
8786ad2antrr 724 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ 𝑆 Fn 𝐼)
88 difssd 4132 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ (𝐼 βˆ– {π‘₯}) βŠ† 𝐼)
89 df-ne 2941 . . . . . . . . . . . . . . . . . 18 (𝑦 β‰  π‘₯ ↔ Β¬ 𝑦 = π‘₯)
90 eldifsn 4790 . . . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ 𝐼 ∧ Β¬ 𝑦 = π‘₯) β†’ 𝑦 ∈ (𝐼 βˆ– {π‘₯}))
9392adantll 712 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ Β¬ 𝑦 = π‘₯) β†’ 𝑦 ∈ (𝐼 βˆ– {π‘₯}))
94 fnfvima 7237 . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . 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 4913 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . 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 19019 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 4566 . . . . . . . . . . 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 7116 . . . . . . . . . 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 495 . . . . . . . . . . . 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 2834 . . . . . . . . . . 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 19887 . . . . . . . . . 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 19886 . . . . . . . . . 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 19788 . . . . . . . . 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 2834 . . . . . . . 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 4194 . . . . . . 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 19881 . . . . . . 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 2835 . . . . . 6 (((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) ∧ π‘₯ ∈ 𝐼) β†’ (πΉβ€˜π‘₯) ∈ { 0 })
112 elsni 4645 . . . . . 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 5248 . . . 4 ((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) β†’ (π‘₯ ∈ 𝐼 ↦ (πΉβ€˜π‘₯)) = (π‘₯ ∈ 𝐼 ↦ 0 ))
1158, 114eqtrd 2772 . . 3 ((πœ‘ ∧ (𝐺 Ξ£g 𝐹) = 0 ) β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 ))
116115ex 413 . 2 (πœ‘ β†’ ((𝐺 Ξ£g 𝐹) = 0 β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 )))
11711gsumz 18719 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )) = 0 )
11846, 22, 117syl2anc 584 . . 3 (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )) = 0 )
119 oveq2 7419 . . . 4 (𝐹 = (π‘₯ ∈ 𝐼 ↦ 0 ) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘₯ ∈ 𝐼 ↦ 0 )))
120119eqeq1d 2734 . . 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  Vcvv 3474   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  ifcif 4528  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∘f cof 7670  Xcixp 8893   finSupp cfsupp 9363  Basecbs 17146  0gc0g 17387   Ξ£g cgsu 17388  Moorecmre 17528  mrClscmrc 17529  ACScacs 17531  Mndcmnd 18627  SubMndcsubmnd 18672  Grpcgrp 18821  -gcsg 18823  SubGrpcsubg 19002  Cntzccntz 19181   DProd cdprd 19865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-seq 13969  df-hash 14293  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-0g 17389  df-gsum 17390  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-submnd 18674  df-grp 18824  df-minusg 18825  df-sbg 18826  df-mulg 18953  df-subg 19005  df-ghm 19092  df-gim 19135  df-cntz 19183  df-oppg 19212  df-cmn 19652  df-dprd 19867
This theorem is referenced by:  dprdf11  19895
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