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Theorem lsmsubg 19616
Description: The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmsubg.p βŠ• = (LSSumβ€˜πΊ)
lsmsubg.z 𝑍 = (Cntzβ€˜πΊ)
Assertion
Ref Expression
lsmsubg ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜πΊ))
Proof of Theorem lsmsubg
Dummy variables π‘Ž 𝑏 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . . 4 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))
2 subgsubm 19110 . . . 4 (𝑇 ∈ (SubGrpβ€˜πΊ) β†’ 𝑇 ∈ (SubMndβ€˜πΊ))
31, 2syl 17 . . 3 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ 𝑇 ∈ (SubMndβ€˜πΊ))
4 simp2 1134 . . . 4 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))
5 subgsubm 19110 . . . 4 (π‘ˆ ∈ (SubGrpβ€˜πΊ) β†’ π‘ˆ ∈ (SubMndβ€˜πΊ))
64, 5syl 17 . . 3 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ π‘ˆ ∈ (SubMndβ€˜πΊ))
7 simp3 1135 . . 3 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))
8 lsmsubg.p . . . 4 βŠ• = (LSSumβ€˜πΊ)
9 lsmsubg.z . . . 4 𝑍 = (Cntzβ€˜πΊ)
108, 9lsmsubm 19615 . . 3 ((𝑇 ∈ (SubMndβ€˜πΊ) ∧ π‘ˆ ∈ (SubMndβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubMndβ€˜πΊ))
113, 6, 7, 10syl3anc 1368 . 2 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubMndβ€˜πΊ))
12 eqid 2728 . . . . . 6 (+gβ€˜πΊ) = (+gβ€˜πΊ)
1312, 8lsmelval 19611 . . . . 5 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (π‘₯ ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘Ž ∈ 𝑇 βˆƒπ‘ ∈ π‘ˆ π‘₯ = (π‘Ž(+gβ€˜πΊ)𝑏)))
14133adant3 1129 . . . 4 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (π‘₯ ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘Ž ∈ 𝑇 βˆƒπ‘ ∈ π‘ˆ π‘₯ = (π‘Ž(+gβ€˜πΊ)𝑏)))
151adantr 479 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))
16 subgrcl 19093 . . . . . . . . . 10 (𝑇 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . 9 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ 𝐺 ∈ Grp)
18 eqid 2728 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
1918subgss 19089 . . . . . . . . . . 11 (𝑇 ∈ (SubGrpβ€˜πΊ) β†’ 𝑇 βŠ† (Baseβ€˜πΊ))
2015, 19syl 17 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ 𝑇 βŠ† (Baseβ€˜πΊ))
21 simprl 769 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ π‘Ž ∈ 𝑇)
2220, 21sseldd 3983 . . . . . . . . 9 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ π‘Ž ∈ (Baseβ€˜πΊ))
234adantr 479 . . . . . . . . . . 11 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))
2418subgss 19089 . . . . . . . . . . 11 (π‘ˆ ∈ (SubGrpβ€˜πΊ) β†’ π‘ˆ βŠ† (Baseβ€˜πΊ))
2523, 24syl 17 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ π‘ˆ βŠ† (Baseβ€˜πΊ))
26 simprr 771 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ 𝑏 ∈ π‘ˆ)
2725, 26sseldd 3983 . . . . . . . . 9 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ 𝑏 ∈ (Baseβ€˜πΊ))
28 eqid 2728 . . . . . . . . . 10 (invgβ€˜πΊ) = (invgβ€˜πΊ)
2918, 12, 28grpinvadd 18981 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ π‘Ž ∈ (Baseβ€˜πΊ) ∧ 𝑏 ∈ (Baseβ€˜πΊ)) β†’ ((invgβ€˜πΊ)β€˜(π‘Ž(+gβ€˜πΊ)𝑏)) = (((invgβ€˜πΊ)β€˜π‘)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘Ž)))
3017, 22, 27, 29syl3anc 1368 . . . . . . . 8 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ ((invgβ€˜πΊ)β€˜(π‘Ž(+gβ€˜πΊ)𝑏)) = (((invgβ€˜πΊ)β€˜π‘)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘Ž)))
317adantr 479 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))
3228subginvcl 19097 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘Ž ∈ 𝑇) β†’ ((invgβ€˜πΊ)β€˜π‘Ž) ∈ 𝑇)
3315, 21, 32syl2anc 582 . . . . . . . . . 10 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ ((invgβ€˜πΊ)β€˜π‘Ž) ∈ 𝑇)
3431, 33sseldd 3983 . . . . . . . . 9 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ ((invgβ€˜πΊ)β€˜π‘Ž) ∈ (π‘β€˜π‘ˆ))
3528subginvcl 19097 . . . . . . . . . 10 ((π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑏 ∈ π‘ˆ) β†’ ((invgβ€˜πΊ)β€˜π‘) ∈ π‘ˆ)
3623, 26, 35syl2anc 582 . . . . . . . . 9 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ ((invgβ€˜πΊ)β€˜π‘) ∈ π‘ˆ)
3712, 9cntzi 19287 . . . . . . . . 9 ((((invgβ€˜πΊ)β€˜π‘Ž) ∈ (π‘β€˜π‘ˆ) ∧ ((invgβ€˜πΊ)β€˜π‘) ∈ π‘ˆ) β†’ (((invgβ€˜πΊ)β€˜π‘Ž)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘)) = (((invgβ€˜πΊ)β€˜π‘)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘Ž)))
3834, 36, 37syl2anc 582 . . . . . . . 8 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ (((invgβ€˜πΊ)β€˜π‘Ž)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘)) = (((invgβ€˜πΊ)β€˜π‘)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘Ž)))
3930, 38eqtr4d 2771 . . . . . . 7 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ ((invgβ€˜πΊ)β€˜(π‘Ž(+gβ€˜πΊ)𝑏)) = (((invgβ€˜πΊ)β€˜π‘Ž)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘)))
4012, 8lsmelvali 19612 . . . . . . . 8 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) ∧ (((invgβ€˜πΊ)β€˜π‘Ž) ∈ 𝑇 ∧ ((invgβ€˜πΊ)β€˜π‘) ∈ π‘ˆ)) β†’ (((invgβ€˜πΊ)β€˜π‘Ž)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘)) ∈ (𝑇 βŠ• π‘ˆ))
4115, 23, 33, 36, 40syl22anc 837 . . . . . . 7 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ (((invgβ€˜πΊ)β€˜π‘Ž)(+gβ€˜πΊ)((invgβ€˜πΊ)β€˜π‘)) ∈ (𝑇 βŠ• π‘ˆ))
4239, 41eqeltrd 2829 . . . . . 6 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ ((invgβ€˜πΊ)β€˜(π‘Ž(+gβ€˜πΊ)𝑏)) ∈ (𝑇 βŠ• π‘ˆ))
43 fveq2 6902 . . . . . . 7 (π‘₯ = (π‘Ž(+gβ€˜πΊ)𝑏) β†’ ((invgβ€˜πΊ)β€˜π‘₯) = ((invgβ€˜πΊ)β€˜(π‘Ž(+gβ€˜πΊ)𝑏)))
4443eleq1d 2814 . . . . . 6 (π‘₯ = (π‘Ž(+gβ€˜πΊ)𝑏) β†’ (((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ) ↔ ((invgβ€˜πΊ)β€˜(π‘Ž(+gβ€˜πΊ)𝑏)) ∈ (𝑇 βŠ• π‘ˆ)))
4542, 44syl5ibrcom 246 . . . . 5 (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) ∧ (π‘Ž ∈ 𝑇 ∧ 𝑏 ∈ π‘ˆ)) β†’ (π‘₯ = (π‘Ž(+gβ€˜πΊ)𝑏) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ)))
4645rexlimdvva 3209 . . . 4 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (βˆƒπ‘Ž ∈ 𝑇 βˆƒπ‘ ∈ π‘ˆ π‘₯ = (π‘Ž(+gβ€˜πΊ)𝑏) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ)))
4714, 46sylbid 239 . . 3 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (π‘₯ ∈ (𝑇 βŠ• π‘ˆ) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ)))
4847ralrimiv 3142 . 2 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ βˆ€π‘₯ ∈ (𝑇 βŠ• π‘ˆ)((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ))
491, 16syl 17 . . 3 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ 𝐺 ∈ Grp)
5028issubg3 19106 . . 3 (𝐺 ∈ Grp β†’ ((𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜πΊ) ↔ ((𝑇 βŠ• π‘ˆ) ∈ (SubMndβ€˜πΊ) ∧ βˆ€π‘₯ ∈ (𝑇 βŠ• π‘ˆ)((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ))))
5149, 50syl 17 . 2 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ ((𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜πΊ) ↔ ((𝑇 βŠ• π‘ˆ) ∈ (SubMndβ€˜πΊ) ∧ βˆ€π‘₯ ∈ (𝑇 βŠ• π‘ˆ)((invgβ€˜πΊ)β€˜π‘₯) ∈ (𝑇 βŠ• π‘ˆ))))
5211, 48, 51mpbir2and 711 1 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067   βŠ† wss 3949  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  SubMndcsubmnd 18746  Grpcgrp 18897  invgcminusg 18898  SubGrpcsubg 19082  Cntzccntz 19273  LSSumclsm 19596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-grp 18900  df-minusg 18901  df-subg 19085  df-cntz 19275  df-lsm 19598
This theorem is referenced by:  pj1ghm  19665  lsmsubg2  19821  dprd2da  20006  dmdprdsplit2lem  20009  dprdsplit  20012
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