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Theorem lsmsubg 19574
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 19075 . . . 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 19075 . . . 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 19573 . . 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 2726 . . . . . 6 (+gβ€˜πΊ) = (+gβ€˜πΊ)
1312, 8lsmelval 19569 . . . . 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 480 . . . . . . . . . 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 19058 . . . . . . . . . 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 2726 . . . . . . . . . . . 12 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
1918subgss 19054 . . . . . . . . . . 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 768 . . . . . . . . . 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 3978 . . . . . . . . 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 480 . . . . . . . . . . 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 19054 . . . . . . . . . . 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 770 . . . . . . . . . 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 3978 . . . . . . . . 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 2726 . . . . . . . . . 10 (invgβ€˜πΊ) = (invgβ€˜πΊ)
2918, 12, 28grpinvadd 18946 . . . . . . . . 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 480 . . . . . . . . . 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 19062 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘Ž ∈ 𝑇) β†’ ((invgβ€˜πΊ)β€˜π‘Ž) ∈ 𝑇)
3315, 21, 32syl2anc 583 . . . . . . . . . 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 3978 . . . . . . . . 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 19062 . . . . . . . . . 10 ((π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑏 ∈ π‘ˆ) β†’ ((invgβ€˜πΊ)β€˜π‘) ∈ π‘ˆ)
3623, 26, 35syl2anc 583 . . . . . . . . 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 19245 . . . . . . . . 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 583 . . . . . . . 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 2769 . . . . . . 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 19570 . . . . . . . 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 836 . . . . . . 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 2827 . . . . . 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 6885 . . . . . . 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 2812 . . . . . 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 3205 . . . 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 3139 . 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 19071 . . 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 710 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  SubMndcsubmnd 18712  Grpcgrp 18863  invgcminusg 18864  SubGrpcsubg 19047  Cntzccntz 19231  LSSumclsm 19554
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-subg 19050  df-cntz 19233  df-lsm 19556
This theorem is referenced by:  pj1ghm  19623  lsmsubg2  19779  dprd2da  19964  dmdprdsplit2lem  19967  dprdsplit  19970
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