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Mirrors > Home > MPE Home > Th. List > cnsubrglem | Structured version Visualization version GIF version |
Description: Lemma for resubdrg 21542 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11216. (Revised by GG, 30-Apr-2025.) |
Ref | Expression |
---|---|
cnsubglem.1 | β’ (π₯ β π΄ β π₯ β β) |
cnsubglem.2 | β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ + π¦) β π΄) |
cnsubglem.3 | β’ (π₯ β π΄ β -π₯ β π΄) |
cnsubrglem.4 | β’ 1 β π΄ |
cnsubrglem.5 | β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ Β· π¦) β π΄) |
Ref | Expression |
---|---|
cnsubrglem | β’ π΄ β (SubRingββfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsubglem.1 | . . 3 β’ (π₯ β π΄ β π₯ β β) | |
2 | cnsubglem.2 | . . 3 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ + π¦) β π΄) | |
3 | cnsubglem.3 | . . 3 β’ (π₯ β π΄ β -π₯ β π΄) | |
4 | cnsubrglem.4 | . . 3 β’ 1 β π΄ | |
5 | 1, 2, 3, 4 | cnsubglem 21350 | . 2 β’ π΄ β (SubGrpββfld) |
6 | cnsubrglem.5 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ Β· π¦) β π΄) | |
7 | 1 | adantr 479 | . . . . . . . 8 β’ ((π₯ β π΄ β§ π¦ β π΄) β π₯ β β) |
8 | 1 | ax-gen 1789 | . . . . . . . . . 10 β’ βπ₯(π₯ β π΄ β π₯ β β) |
9 | eleq1 2813 | . . . . . . . . . . . 12 β’ (π₯ = π¦ β (π₯ β π΄ β π¦ β π΄)) | |
10 | eleq1 2813 | . . . . . . . . . . . 12 β’ (π₯ = π¦ β (π₯ β β β π¦ β β)) | |
11 | 9, 10 | imbi12d 343 | . . . . . . . . . . 11 β’ (π₯ = π¦ β ((π₯ β π΄ β π₯ β β) β (π¦ β π΄ β π¦ β β))) |
12 | 11 | spvv 1992 | . . . . . . . . . 10 β’ (βπ₯(π₯ β π΄ β π₯ β β) β (π¦ β π΄ β π¦ β β)) |
13 | 8, 12 | ax-mp 5 | . . . . . . . . 9 β’ (π¦ β π΄ β π¦ β β) |
14 | 13 | adantl 480 | . . . . . . . 8 β’ ((π₯ β π΄ β§ π¦ β π΄) β π¦ β β) |
15 | 7, 14 | jca 510 | . . . . . . 7 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ β β β§ π¦ β β)) |
16 | ovmpot 7578 | . . . . . . 7 β’ ((π₯ β β β§ π¦ β β) β (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) = (π₯ Β· π¦)) | |
17 | 15, 16 | syl 17 | . . . . . 6 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) = (π₯ Β· π¦)) |
18 | 17 | eqcomd 2731 | . . . . 5 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ Β· π¦) = (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦)) |
19 | 18 | eleq1d 2810 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β ((π₯ Β· π¦) β π΄ β (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) β π΄)) |
20 | 6, 19 | mpbid 231 | . . 3 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) β π΄) |
21 | 20 | rgen2 3188 | . 2 β’ βπ₯ β π΄ βπ¦ β π΄ (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) β π΄ |
22 | cnring 21320 | . . 3 β’ βfld β Ring | |
23 | cnfldbas 21285 | . . . 4 β’ β = (Baseββfld) | |
24 | cnfld1 21323 | . . . 4 β’ 1 = (1rββfld) | |
25 | mpocnfldmul 21288 | . . . 4 β’ (π’ β β, π£ β β β¦ (π’ Β· π£)) = (.rββfld) | |
26 | 23, 24, 25 | issubrg2 20533 | . . 3 β’ (βfld β Ring β (π΄ β (SubRingββfld) β (π΄ β (SubGrpββfld) β§ 1 β π΄ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) β π΄))) |
27 | 22, 26 | ax-mp 5 | . 2 β’ (π΄ β (SubRingββfld) β (π΄ β (SubGrpββfld) β§ 1 β π΄ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) β π΄)) |
28 | 5, 4, 21, 27 | mpbir3an 1338 | 1 β’ π΄ β (SubRingββfld) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 βwal 1531 = wceq 1533 β wcel 2098 βwral 3051 βcfv 6542 (class class class)co 7415 β cmpo 7417 βcc 11134 1c1 11137 + caddc 11139 Β· cmul 11141 -cneg 11473 SubGrpcsubg 19077 Ringcrg 20175 SubRingcsubrg 20508 βfldccnfld 21281 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-subg 19080 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-subrng 20485 df-subrg 20510 df-cnfld 21282 |
This theorem is referenced by: cnsubdrglem 21353 zsubrg 21355 gzsubrg 21356 cnstrcvs 25084 cncvs 25088 |
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