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Mirrors > Home > MPE Home > Th. List > subrg1cl | Structured version Visualization version GIF version |
Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrg1cl.a | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
subrg1cl | β’ (π΄ β (SubRingβπ ) β 1 β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
2 | subrg1cl.a | . . . 4 β’ 1 = (1rβπ ) | |
3 | 1, 2 | issubrg 20462 | . . 3 β’ (π΄ β (SubRingβπ ) β ((π β Ring β§ (π βΎs π΄) β Ring) β§ (π΄ β (Baseβπ ) β§ 1 β π΄))) |
4 | 3 | simprbi 496 | . 2 β’ (π΄ β (SubRingβπ ) β (π΄ β (Baseβπ ) β§ 1 β π΄)) |
5 | 4 | simprd 495 | 1 β’ (π΄ β (SubRingβπ ) β 1 β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wss 3948 βcfv 6543 (class class class)co 7412 Basecbs 17149 βΎs cress 17178 1rcur 20076 Ringcrg 20128 SubRingcsubrg 20458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 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-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-subrg 20460 |
This theorem is referenced by: subrg1 20473 subrgsubm 20476 issubrg2 20483 subrgint 20486 subsubrg 20489 primefld1cl 20567 zsssubrg 21204 issubassa2 21666 subrgpsr 21759 mplassa 21801 mplbas2 21817 ply1assa 21943 isclmp 24845 taylply2 26117 0ringsubrg 32650 subrgchr 32657 fldgensdrg 32675 primefldgen1 32682 asclply1subcl 32935 drgextlsp 32969 evls1fldgencl 33034 evls1maprhm 33049 ply1annnr 33054 algextdeglem4 33066 evlsmaprhm 41445 cnsrexpcl 42210 rngunsnply 42218 |
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