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Theorem lsmsubg 19516
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 1136 . . . 4 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))
2 subgsubm 19022 . . . 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 1137 . . . 4 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))
5 subgsubm 19022 . . . 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 1138 . . 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 19515 . . 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 1371 . 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 2732 . . . . . 6 (+gβ€˜πΊ) = (+gβ€˜πΊ)
1312, 8lsmelval 19511 . . . . 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 1132 . . . 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 481 . . . . . . . . . 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 19005 . . . . . . . . . 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 2732 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
1918subgss 19001 . . . . . . . . . . 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 3982 . . . . . . . . 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 481 . . . . . . . . . . 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 19001 . . . . . . . . . . 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 3982 . . . . . . . . 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 2732 . . . . . . . . . 10 (invgβ€˜πΊ) = (invgβ€˜πΊ)
2918, 12, 28grpinvadd 18897 . . . . . . . . 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 1371 . . . . . . . 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 481 . . . . . . . . . 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 19009 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘Ž ∈ 𝑇) β†’ ((invgβ€˜πΊ)β€˜π‘Ž) ∈ 𝑇)
3315, 21, 32syl2anc 584 . . . . . . . . . 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 3982 . . . . . . . . 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 19009 . . . . . . . . . 10 ((π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑏 ∈ π‘ˆ) β†’ ((invgβ€˜πΊ)β€˜π‘) ∈ π‘ˆ)
3623, 26, 35syl2anc 584 . . . . . . . . 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 19187 . . . . . . . . 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 584 . . . . . . . 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 2775 . . . . . . 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 19512 . . . . . . . 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 2833 . . . . . 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 6888 . . . . . . 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 2818 . . . . . 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 3211 . . . 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 3145 . 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 19018 . . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3947  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  SubMndcsubmnd 18666  Grpcgrp 18815  invgcminusg 18816  SubGrpcsubg 18994  Cntzccntz 19173  LSSumclsm 19496
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-grp 18818  df-minusg 18819  df-subg 18997  df-cntz 19175  df-lsm 19498
This theorem is referenced by:  pj1ghm  19565  lsmsubg2  19721  dprd2da  19906  dmdprdsplit2lem  19909  dprdsplit  19912
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