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Theorem subgintm 13058
Description: The intersection of an inhabited collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
Assertion
Ref Expression
subgintm ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 ∈ (SubGrpβ€˜πΊ))
Distinct variable groups:   𝑀,𝐺   𝑀,𝑆
Proof of Theorem subgintm
Dummy variables π‘₯ 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 intssunim 3867 . . . 4 (βˆƒπ‘€ 𝑀 ∈ 𝑆 β†’ ∩ 𝑆 βŠ† βˆͺ 𝑆)
21adantl 277 . . 3 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 βŠ† βˆͺ 𝑆)
3 ssel2 3151 . . . . . . 7 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ 𝑔 ∈ 𝑆) β†’ 𝑔 ∈ (SubGrpβ€˜πΊ))
43adantlr 477 . . . . . 6 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ 𝑔 ∈ 𝑆) β†’ 𝑔 ∈ (SubGrpβ€˜πΊ))
5 eqid 2177 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
65subgss 13034 . . . . . 6 (𝑔 ∈ (SubGrpβ€˜πΊ) β†’ 𝑔 βŠ† (Baseβ€˜πΊ))
74, 6syl 14 . . . . 5 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ 𝑔 ∈ 𝑆) β†’ 𝑔 βŠ† (Baseβ€˜πΊ))
87ralrimiva 2550 . . . 4 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆ€π‘” ∈ 𝑆 𝑔 βŠ† (Baseβ€˜πΊ))
9 unissb 3840 . . . 4 (βˆͺ 𝑆 βŠ† (Baseβ€˜πΊ) ↔ βˆ€π‘” ∈ 𝑆 𝑔 βŠ† (Baseβ€˜πΊ))
108, 9sylibr 134 . . 3 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆͺ 𝑆 βŠ† (Baseβ€˜πΊ))
112, 10sstrd 3166 . 2 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 βŠ† (Baseβ€˜πΊ))
12 eqid 2177 . . . . . . 7 (0gβ€˜πΊ) = (0gβ€˜πΊ)
1312subg0cl 13042 . . . . . 6 (𝑔 ∈ (SubGrpβ€˜πΊ) β†’ (0gβ€˜πΊ) ∈ 𝑔)
144, 13syl 14 . . . . 5 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ 𝑔 ∈ 𝑆) β†’ (0gβ€˜πΊ) ∈ 𝑔)
1514ralrimiva 2550 . . . 4 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆ€π‘” ∈ 𝑆 (0gβ€˜πΊ) ∈ 𝑔)
16 ssel 3150 . . . . . . . 8 (𝑆 βŠ† (SubGrpβ€˜πΊ) β†’ (𝑀 ∈ 𝑆 β†’ 𝑀 ∈ (SubGrpβ€˜πΊ)))
1716eximdv 1880 . . . . . . 7 (𝑆 βŠ† (SubGrpβ€˜πΊ) β†’ (βˆƒπ‘€ 𝑀 ∈ 𝑆 β†’ βˆƒπ‘€ 𝑀 ∈ (SubGrpβ€˜πΊ)))
1817imp 124 . . . . . 6 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆƒπ‘€ 𝑀 ∈ (SubGrpβ€˜πΊ))
19 subgrcl 13039 . . . . . . 7 (𝑀 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
2019exlimiv 1598 . . . . . 6 (βˆƒπ‘€ 𝑀 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
2118, 20syl 14 . . . . 5 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ 𝐺 ∈ Grp)
225, 12grpidcl 12904 . . . . 5 (𝐺 ∈ Grp β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
23 elintg 3853 . . . . 5 ((0gβ€˜πΊ) ∈ (Baseβ€˜πΊ) β†’ ((0gβ€˜πΊ) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 (0gβ€˜πΊ) ∈ 𝑔))
2421, 22, 233syl 17 . . . 4 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ((0gβ€˜πΊ) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 (0gβ€˜πΊ) ∈ 𝑔))
2515, 24mpbird 167 . . 3 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ (0gβ€˜πΊ) ∈ ∩ 𝑆)
26 elex2 2754 . . 3 ((0gβ€˜πΊ) ∈ ∩ 𝑆 β†’ βˆƒπ‘€ 𝑀 ∈ ∩ 𝑆)
2725, 26syl 14 . 2 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆƒπ‘€ 𝑀 ∈ ∩ 𝑆)
284adantlr 477 . . . . . . . . 9 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) β†’ 𝑔 ∈ (SubGrpβ€˜πΊ))
29 simprl 529 . . . . . . . . . 10 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ π‘₯ ∈ ∩ 𝑆)
30 elinti 3854 . . . . . . . . . . 11 (π‘₯ ∈ ∩ 𝑆 β†’ (𝑔 ∈ 𝑆 β†’ π‘₯ ∈ 𝑔))
3130imp 124 . . . . . . . . . 10 ((π‘₯ ∈ ∩ 𝑆 ∧ 𝑔 ∈ 𝑆) β†’ π‘₯ ∈ 𝑔)
3229, 31sylan 283 . . . . . . . . 9 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) β†’ π‘₯ ∈ 𝑔)
33 simprr 531 . . . . . . . . . 10 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ 𝑦 ∈ ∩ 𝑆)
34 elinti 3854 . . . . . . . . . . 11 (𝑦 ∈ ∩ 𝑆 β†’ (𝑔 ∈ 𝑆 β†’ 𝑦 ∈ 𝑔))
3534imp 124 . . . . . . . . . 10 ((𝑦 ∈ ∩ 𝑆 ∧ 𝑔 ∈ 𝑆) β†’ 𝑦 ∈ 𝑔)
3633, 35sylan 283 . . . . . . . . 9 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) β†’ 𝑦 ∈ 𝑔)
37 eqid 2177 . . . . . . . . . 10 (+gβ€˜πΊ) = (+gβ€˜πΊ)
3837subgcl 13044 . . . . . . . . 9 ((𝑔 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ 𝑔 ∧ 𝑦 ∈ 𝑔) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑔)
3928, 32, 36, 38syl3anc 1238 . . . . . . . 8 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) ∧ 𝑔 ∈ 𝑆) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑔)
4039ralrimiva 2550 . . . . . . 7 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ βˆ€π‘” ∈ 𝑆 (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑔)
41 vex 2741 . . . . . . . . . . 11 π‘₯ ∈ V
4241a1i 9 . . . . . . . . . 10 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ π‘₯ ∈ V)
43 plusgslid 12571 . . . . . . . . . . . 12 (+g = Slot (+gβ€˜ndx) ∧ (+gβ€˜ndx) ∈ β„•)
4443slotex 12489 . . . . . . . . . . 11 (𝐺 ∈ Grp β†’ (+gβ€˜πΊ) ∈ V)
4518, 20, 443syl 17 . . . . . . . . . 10 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ (+gβ€˜πΊ) ∈ V)
46 vex 2741 . . . . . . . . . . 11 𝑦 ∈ V
4746a1i 9 . . . . . . . . . 10 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ 𝑦 ∈ V)
48 ovexg 5909 . . . . . . . . . 10 ((π‘₯ ∈ V ∧ (+gβ€˜πΊ) ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ V)
4942, 45, 47, 48syl3anc 1238 . . . . . . . . 9 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ V)
50 elintg 3853 . . . . . . . . 9 ((π‘₯(+gβ€˜πΊ)𝑦) ∈ V β†’ ((π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑔))
5149, 50syl 14 . . . . . . . 8 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ((π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑔))
5251adantr 276 . . . . . . 7 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ ((π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 (π‘₯(+gβ€˜πΊ)𝑦) ∈ 𝑔))
5340, 52mpbird 167 . . . . . 6 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ (π‘₯ ∈ ∩ 𝑆 ∧ 𝑦 ∈ ∩ 𝑆)) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆)
5453anassrs 400 . . . . 5 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) ∧ 𝑦 ∈ ∩ 𝑆) β†’ (π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆)
5554ralrimiva 2550 . . . 4 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆)
564adantlr 477 . . . . . . 7 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) ∧ 𝑔 ∈ 𝑆) β†’ 𝑔 ∈ (SubGrpβ€˜πΊ))
5731adantll 476 . . . . . . 7 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) ∧ 𝑔 ∈ 𝑆) β†’ π‘₯ ∈ 𝑔)
58 eqid 2177 . . . . . . . 8 (invgβ€˜πΊ) = (invgβ€˜πΊ)
5958subginvcl 13043 . . . . . . 7 ((𝑔 ∈ (SubGrpβ€˜πΊ) ∧ π‘₯ ∈ 𝑔) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ 𝑔)
6056, 57, 59syl2anc 411 . . . . . 6 ((((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) ∧ 𝑔 ∈ 𝑆) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ 𝑔)
6160ralrimiva 2550 . . . . 5 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ βˆ€π‘” ∈ 𝑆 ((invgβ€˜πΊ)β€˜π‘₯) ∈ 𝑔)
6221adantr 276 . . . . . . 7 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ 𝐺 ∈ Grp)
6311sselda 3156 . . . . . . 7 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
645, 58grpinvcl 12921 . . . . . . 7 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ (Baseβ€˜πΊ))
6562, 63, 64syl2anc 411 . . . . . 6 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ (Baseβ€˜πΊ))
66 elintg 3853 . . . . . 6 (((invgβ€˜πΊ)β€˜π‘₯) ∈ (Baseβ€˜πΊ) β†’ (((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 ((invgβ€˜πΊ)β€˜π‘₯) ∈ 𝑔))
6765, 66syl 14 . . . . 5 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ (((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆 ↔ βˆ€π‘” ∈ 𝑆 ((invgβ€˜πΊ)β€˜π‘₯) ∈ 𝑔))
6861, 67mpbird 167 . . . 4 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ ((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆)
6955, 68jca 306 . . 3 (((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) ∧ π‘₯ ∈ ∩ 𝑆) β†’ (βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ∧ ((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆))
7069ralrimiva 2550 . 2 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ βˆ€π‘₯ ∈ ∩ 𝑆(βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ∧ ((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆))
715, 37, 58issubg2m 13049 . . 3 (𝐺 ∈ Grp β†’ (∩ 𝑆 ∈ (SubGrpβ€˜πΊ) ↔ (∩ 𝑆 βŠ† (Baseβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ ∩ 𝑆 ∧ βˆ€π‘₯ ∈ ∩ 𝑆(βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ∧ ((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆))))
7218, 20, 713syl 17 . 2 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ (∩ 𝑆 ∈ (SubGrpβ€˜πΊ) ↔ (∩ 𝑆 βŠ† (Baseβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ ∩ 𝑆 ∧ βˆ€π‘₯ ∈ ∩ 𝑆(βˆ€π‘¦ ∈ ∩ 𝑆(π‘₯(+gβ€˜πΊ)𝑦) ∈ ∩ 𝑆 ∧ ((invgβ€˜πΊ)β€˜π‘₯) ∈ ∩ 𝑆))))
7311, 27, 70, 72mpbir3and 1180 1 ((𝑆 βŠ† (SubGrpβ€˜πΊ) ∧ βˆƒπ‘€ 𝑀 ∈ 𝑆) β†’ ∩ 𝑆 ∈ (SubGrpβ€˜πΊ))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  Vcvv 2738   βŠ† wss 3130  βˆͺ cuni 3810  βˆ© cint 3845  β€˜cfv 5217  (class class class)co 5875  Basecbs 12462  +gcplusg 12536  0gc0g 12705  Grpcgrp 12877  invgcminusg 12878  SubGrpcsubg 13027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-iress 12470  df-plusg 12549  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-subg 13030
This theorem is referenced by:  subrgintm  13364
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