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Theorem dprdsn 19901
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 19006 . . . 4 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
54adantl 483 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ 𝐺 ∈ Grp)
6 snex 5431 . . . 4 {𝐴} ∈ V
76a1i 11 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {𝐴} ∈ V)
8 f1osng 6872 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {⟨𝐴, π‘†βŸ©}:{𝐴}–1-1-ontoβ†’{𝑆})
9 f1of 6831 . . . . 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 4812 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ {𝑆} βŠ† (SubGrpβ€˜πΊ))
1310, 12fssd 6733 . . 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 4645 . . . . . 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 4645 . . . . . 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 19036 . . . . . . . 8 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ (ACSβ€˜(Baseβ€˜πΊ)))
26 acsmre 17593 . . . . . . . 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 4640 . . . . . . . . . . . . . 14 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ {π‘₯} = {𝐴})
3029difeq2d 4122 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({𝐴} βˆ– {π‘₯}) = ({𝐴} βˆ– {𝐴}))
31 difid 4370 . . . . . . . . . . . . 13 ({𝐴} βˆ– {𝐴}) = βˆ…
3230, 31eqtrdi 2789 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ ({𝐴} βˆ– {π‘₯}) = βˆ…)
3332imaeq2d 6058 . . . . . . . . . . 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 6074 . . . . . . . . . . 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 4922 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) = βˆͺ βˆ…)
37 uni0 4939 . . . . . . . . 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 4396 . . . . . . . . 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 4020 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ βˆͺ ({⟨𝐴, π‘†βŸ©} β€œ ({𝐴} βˆ– {π‘₯})) βŠ† {(0gβ€˜πΊ)})
4220subg 19026 . . . . . . . 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 17561 . . . . . . 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 19009 . . . . . . . . 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 6893 . . . . . . . . 9 (π‘₯ ∈ {𝐴} β†’ ({⟨𝐴, π‘†βŸ©}β€˜π‘₯) = ({⟨𝐴, π‘†βŸ©}β€˜π΄))
49 fvsng 7175 . . . . . . . . 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 4812 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ {𝐴}) β†’ {(0gβ€˜πΊ)} βŠ† ({⟨𝐴, π‘†βŸ©}β€˜π‘₯))
5345, 52sstrd 3992 . . . . 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 4215 . . . . 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 4020 . . 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 19864 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ 𝐺dom DProd {⟨𝐴, π‘†βŸ©})
583dprdspan 19892 . . . 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 6218 . . . . . . . 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 4922 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝑆 ∈ (SubGrpβ€˜πΊ)) β†’ βˆͺ ran {⟨𝐴, π‘†βŸ©} = βˆͺ {𝑆})
63 unisng 4929 . . . . . . 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 6893 . . . 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 17554 . . . . 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 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148  dom cdm 5676  ran crn 5677   β€œ cima 5679  βŸΆwf 6537  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  (class class class)co 7406  Basecbs 17141  0gc0g 17382  Moorecmre 17523  mrClscmrc 17524  ACScacs 17526  Grpcgrp 18816  SubGrpcsubg 18995  Cntzccntz 19174   DProd cdprd 19858
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
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 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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-supp 8144  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-0g 17384  df-gsum 17385  df-mre 17527  df-mrc 17528  df-acs 17530  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-mhm 18668  df-submnd 18669  df-grp 18819  df-minusg 18820  df-sbg 18821  df-mulg 18946  df-subg 18998  df-ghm 19085  df-gim 19128  df-cntz 19176  df-oppg 19205  df-cmn 19645  df-dprd 19860
This theorem is referenced by:  dprd2da  19907  dmdprdpr  19914  dprdpr  19915  dpjlem  19916  pgpfaclem1  19946
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