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Theorem dprdsn 19906
Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdsn ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (𝐺dom DProd {⟨𝐴, π‘†βŸ©} ∧ (𝐺 DProd {⟨𝐴, π‘†βŸ©}) = 𝑆))
Proof of Theorem dprdsn
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Cntzβ€˜πΊ) = (Cntzβ€˜πΊ)
2 eqid 2733 . . 3 (0gβ€˜πΊ) = (0gβ€˜πΊ)
3 eqid 2733 . . 3 (mrClsβ€˜(SubGrpβ€˜πΊ)) = (mrClsβ€˜(SubGrpβ€˜πΊ))
4 subgrcl 19011 . . . 4 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
54adantl 483 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ 𝐺 ∈ Grp)
6 snex 5432 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {𝐴} ∈ V)
8 f1osng 6875 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {⟨𝐴, π‘†βŸ©}:{𝐴}–1-1-ontoβ†’{𝑆})
9 f1of 6834 . . . . 5 ({⟨𝐴, π‘†βŸ©}:{𝐴}–1-1-ontoβ†’{𝑆} β†’ {⟨𝐴, π‘†βŸ©}:{𝐴}⟢{𝑆})
108, 9syl 17 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {⟨𝐴, π‘†βŸ©}:{𝐴}⟢{𝑆})
11 simpr 486 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
1211snssd 4813 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {𝑆} βŠ† (SubGrpβ€˜πΊ))
1310, 12fssd 6736 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {⟨𝐴, π‘†βŸ©}:{𝐴}⟢(SubGrpβ€˜πΊ))
14 simpr1 1195 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ π‘₯ ∈ {𝐴})
15 elsni 4646 . . . . . 6 (π‘₯ ∈ {𝐴} β†’ π‘₯ = 𝐴)
1614, 15syl 17 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ π‘₯ = 𝐴)
17 simpr2 1196 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ 𝑦 ∈ {𝐴})
18 elsni 4646 . . . . . 6 (𝑦 ∈ {𝐴} β†’ 𝑦 = 𝐴)
1917, 18syl 17 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ 𝑦 = 𝐴)
2016, 19eqtr4d 2776 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ π‘₯ = 𝑦)
21 simpr3 1197 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ π‘₯ β‰  𝑦)
2220, 21pm2.21ddne 3027 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ (π‘₯ ∈ {𝐴} ∧ 𝑦 ∈ {𝐴} ∧ π‘₯ β‰  𝑦)) β†’ ({⟨𝐴, π‘†βŸ©}β€˜π‘₯) βŠ† ((Cntzβ€˜πΊ)β€˜({⟨𝐴, π‘†βŸ©}β€˜π‘¦)))
235adantr 482 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ 𝐺 ∈ Grp)
24 eqid 2733 . . . . . . . . 9 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
2524subgacs 19041 . . . . . . . 8 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ (ACSβ€˜(Baseβ€˜πΊ)))
26 acsmre 17596 . . . . . . . 8 ((SubGrpβ€˜πΊ) ∈ (ACSβ€˜(Baseβ€˜πΊ)) β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)))
2723, 25, 263syl 18 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)))
2815adantl 483 . . . . . . . . . . . . . . 15 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ π‘₯ = 𝐴)
2928sneqd 4641 . . . . . . . . . . . . . 14 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ {π‘₯} = {𝐴})
3029difeq2d 4123 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({𝐴} βˆ– {π‘₯}) = ({𝐴} βˆ– {𝐴}))
31 difid 4371 . . . . . . . . . . . . 13 ({𝐴} βˆ– {𝐴}) = βˆ…
3230, 31eqtrdi 2789 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({𝐴} βˆ– {π‘₯}) = βˆ…)
3332imaeq2d 6060 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) = ({⟨𝐴, π‘†βŸ©} β€œ βˆ…))
34 ima0 6077 . . . . . . . . . . 11 ({⟨𝐴, π‘†βŸ©} β€œ βˆ…) = βˆ…
3533, 34eqtrdi 2789 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) = βˆ…)
3635unieqd 4923 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) = βˆͺ βˆ…)
37 uni0 4940 . . . . . . . . 9 βˆͺ βˆ… = βˆ…
3836, 37eqtrdi 2789 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) = βˆ…)
39 0ss 4397 . . . . . . . . 9 βˆ… βŠ† {(0gβ€˜πΊ)}
4039a1i 11 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ βˆ… βŠ† {(0gβ€˜πΊ)})
4138, 40eqsstrd 4021 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) βŠ† {(0gβ€˜πΊ)})
4220subg 19031 . . . . . . . 8 (𝐺 ∈ Grp β†’ {(0gβ€˜πΊ)} ∈ (SubGrpβ€˜πΊ))
4323, 42syl 17 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ {(0gβ€˜πΊ)} ∈ (SubGrpβ€˜πΊ))
443mrcsscl 17564 . . . . . . 7 (((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)) ∧ βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) βŠ† {(0gβ€˜πΊ)} ∧ {(0gβ€˜πΊ)} ∈ (SubGrpβ€˜πΊ)) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯}))) βŠ† {(0gβ€˜πΊ)})
4527, 41, 43, 44syl3anc 1372 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯}))) βŠ† {(0gβ€˜πΊ)})
462subg0cl 19014 . . . . . . . . 9 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ (0gβ€˜πΊ) ∈ 𝑆)
4746ad2antlr 726 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ (0gβ€˜πΊ) ∈ 𝑆)
4815fveq2d 6896 . . . . . . . . 9 (π‘₯ ∈ {𝐴} β†’ ({⟨𝐴, π‘†βŸ©}β€˜π‘₯) = ({⟨𝐴, π‘†βŸ©}β€˜π΄))
49 fvsng 7178 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ({⟨𝐴, π‘†βŸ©}β€˜π΄) = 𝑆)
5048, 49sylan9eqr 2795 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({⟨𝐴, π‘†βŸ©}β€˜π‘₯) = 𝑆)
5147, 50eleqtrrd 2837 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ (0gβ€˜πΊ) ∈ ({⟨𝐴, π‘†βŸ©}β€˜π‘₯))
5251snssd 4813 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ {(0gβ€˜πΊ)} βŠ† ({⟨𝐴, π‘†βŸ©}β€˜π‘₯))
5345, 52sstrd 3993 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯}))) βŠ† ({⟨𝐴, π‘†βŸ©}β€˜π‘₯))
54 sseqin2 4216 . . . . 5 (((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯}))) βŠ† ({⟨𝐴, π‘†βŸ©}β€˜π‘₯) ↔ (({⟨𝐴, π‘†βŸ©}β€˜π‘₯) ∩ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})))) = ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯}))))
5553, 54sylib 217 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ (({⟨𝐴, π‘†βŸ©}β€˜π‘₯) ∩ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})))) = ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯}))))
5655, 45eqsstrd 4021 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ (({⟨𝐴, π‘†βŸ©}β€˜π‘₯) ∩ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})))) βŠ† {(0gβ€˜πΊ)})
571, 2, 3, 5, 7, 13, 22, 56dmdprdd 19869 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ 𝐺dom DProd {⟨𝐴, π‘†βŸ©})
583dprdspan 19897 . . . 4 (𝐺dom DProd {⟨𝐴, π‘†βŸ©} β†’ (𝐺 DProd {⟨𝐴, π‘†βŸ©}) = ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ran {⟨𝐴, π‘†βŸ©}))
5957, 58syl 17 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (𝐺 DProd {⟨𝐴, π‘†βŸ©}) = ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ran {⟨𝐴, π‘†βŸ©}))
60 rnsnopg 6221 . . . . . . . 8 (𝐴 ∈ 𝑉 β†’ ran {⟨𝐴, π‘†βŸ©} = {𝑆})
6160adantr 482 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ran {⟨𝐴, π‘†βŸ©} = {𝑆})
6261unieqd 4923 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ βˆͺ ran {⟨𝐴, π‘†βŸ©} = βˆͺ {𝑆})
63 unisng 4930 . . . . . . 7 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ βˆͺ {𝑆} = 𝑆)
6463adantl 483 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ βˆͺ {𝑆} = 𝑆)
6562, 64eqtrd 2773 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ βˆͺ ran {⟨𝐴, π‘†βŸ©} = 𝑆)
6665fveq2d 6896 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ran {⟨𝐴, π‘†βŸ©}) = ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜π‘†))
675, 25, 263syl 18 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)))
683mrcid 17557 . . . . 5 (((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜(Baseβ€˜πΊ)) ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜π‘†) = 𝑆)
6967, 68sylancom 589 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜π‘†) = 𝑆)
7066, 69eqtrd 2773 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ ((mrClsβ€˜(SubGrpβ€˜πΊ))β€˜βˆͺ ran {⟨𝐴, π‘†βŸ©}) = 𝑆)
7159, 70eqtrd 2773 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (𝐺 DProd {⟨𝐴, π‘†βŸ©}) = 𝑆)
7257, 71jca 513 1 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ (𝐺dom DProd {⟨𝐴, π‘†βŸ©} ∧ (𝐺 DProd {⟨𝐴, π‘†βŸ©}) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βŸ¨cop 4635  βˆͺ cuni 4909   class class class wbr 5149  dom cdm 5677  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  0gc0g 17385  Moorecmre 17526  mrClscmrc 17527  ACScacs 17529  Grpcgrp 18819  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:  dprd2da  19912  dmdprdpr  19919  dprdpr  19920  dpjlem  19921  pgpfaclem1  19951
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