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Mirrors > Home > MPE Home > Th. List > issubrgd | Structured version Visualization version GIF version |
Description: Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) |
Ref | Expression |
---|---|
issubrgd.s | β’ (π β π = (πΌ βΎs π·)) |
issubrgd.z | β’ (π β 0 = (0gβπΌ)) |
issubrgd.p | β’ (π β + = (+gβπΌ)) |
issubrgd.ss | β’ (π β π· β (BaseβπΌ)) |
issubrgd.zcl | β’ (π β 0 β π·) |
issubrgd.acl | β’ ((π β§ π₯ β π· β§ π¦ β π·) β (π₯ + π¦) β π·) |
issubrgd.ncl | β’ ((π β§ π₯ β π·) β ((invgβπΌ)βπ₯) β π·) |
issubrgd.o | β’ (π β 1 = (1rβπΌ)) |
issubrgd.t | β’ (π β Β· = (.rβπΌ)) |
issubrgd.ocl | β’ (π β 1 β π·) |
issubrgd.tcl | β’ ((π β§ π₯ β π· β§ π¦ β π·) β (π₯ Β· π¦) β π·) |
issubrgd.g | β’ (π β πΌ β Ring) |
Ref | Expression |
---|---|
issubrgd | β’ (π β π· β (SubRingβπΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubrgd.s | . . 3 β’ (π β π = (πΌ βΎs π·)) | |
2 | issubrgd.z | . . 3 β’ (π β 0 = (0gβπΌ)) | |
3 | issubrgd.p | . . 3 β’ (π β + = (+gβπΌ)) | |
4 | issubrgd.ss | . . 3 β’ (π β π· β (BaseβπΌ)) | |
5 | issubrgd.zcl | . . 3 β’ (π β 0 β π·) | |
6 | issubrgd.acl | . . 3 β’ ((π β§ π₯ β π· β§ π¦ β π·) β (π₯ + π¦) β π·) | |
7 | issubrgd.ncl | . . 3 β’ ((π β§ π₯ β π·) β ((invgβπΌ)βπ₯) β π·) | |
8 | issubrgd.g | . . . 4 β’ (π β πΌ β Ring) | |
9 | ringgrp 20139 | . . . 4 β’ (πΌ β Ring β πΌ β Grp) | |
10 | 8, 9 | syl 17 | . . 3 β’ (π β πΌ β Grp) |
11 | 1, 2, 3, 4, 5, 6, 7, 10 | issubgrpd2 19065 | . 2 β’ (π β π· β (SubGrpβπΌ)) |
12 | issubrgd.o | . . 3 β’ (π β 1 = (1rβπΌ)) | |
13 | issubrgd.ocl | . . 3 β’ (π β 1 β π·) | |
14 | 12, 13 | eqeltrrd 2833 | . 2 β’ (π β (1rβπΌ) β π·) |
15 | issubrgd.t | . . . . 5 β’ (π β Β· = (.rβπΌ)) | |
16 | 15 | oveqdr 7440 | . . . 4 β’ ((π β§ (π₯ β π· β§ π¦ β π·)) β (π₯ Β· π¦) = (π₯(.rβπΌ)π¦)) |
17 | issubrgd.tcl | . . . . 5 β’ ((π β§ π₯ β π· β§ π¦ β π·) β (π₯ Β· π¦) β π·) | |
18 | 17 | 3expb 1119 | . . . 4 β’ ((π β§ (π₯ β π· β§ π¦ β π·)) β (π₯ Β· π¦) β π·) |
19 | 16, 18 | eqeltrrd 2833 | . . 3 β’ ((π β§ (π₯ β π· β§ π¦ β π·)) β (π₯(.rβπΌ)π¦) β π·) |
20 | 19 | ralrimivva 3199 | . 2 β’ (π β βπ₯ β π· βπ¦ β π· (π₯(.rβπΌ)π¦) β π·) |
21 | eqid 2731 | . . . 4 β’ (BaseβπΌ) = (BaseβπΌ) | |
22 | eqid 2731 | . . . 4 β’ (1rβπΌ) = (1rβπΌ) | |
23 | eqid 2731 | . . . 4 β’ (.rβπΌ) = (.rβπΌ) | |
24 | 21, 22, 23 | issubrg2 20490 | . . 3 β’ (πΌ β Ring β (π· β (SubRingβπΌ) β (π· β (SubGrpβπΌ) β§ (1rβπΌ) β π· β§ βπ₯ β π· βπ¦ β π· (π₯(.rβπΌ)π¦) β π·))) |
25 | 8, 24 | syl 17 | . 2 β’ (π β (π· β (SubRingβπΌ) β (π· β (SubGrpβπΌ) β§ (1rβπΌ) β π· β§ βπ₯ β π· βπ¦ β π· (π₯(.rβπΌ)π¦) β π·))) |
26 | 11, 14, 20, 25 | mpbir3and 1341 | 1 β’ (π β π· β (SubRingβπΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3060 β wss 3948 βcfv 6543 (class class class)co 7412 Basecbs 17151 βΎs cress 17180 +gcplusg 17204 .rcmulr 17205 0gc0g 17392 Grpcgrp 18861 invgcminusg 18862 SubGrpcsubg 19043 1rcur 20082 Ringcrg 20134 SubRingcsubrg 20465 |
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 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 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-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-subrng 20442 df-subrg 20467 |
This theorem is referenced by: rngunsnply 42381 |
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