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Mirrors > Home > MPE Home > Th. List > cnsubrglem | Structured version Visualization version GIF version |
Description: Lemma for resubdrg 21501 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11192. (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 21309 | . 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 480 | . . . . . . . 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 2815 | . . . . . . . . . . . 12 β’ (π₯ = π¦ β (π₯ β π΄ β π¦ β π΄)) | |
10 | eleq1 2815 | . . . . . . . . . . . 12 β’ (π₯ = π¦ β (π₯ β β β π¦ β β)) | |
11 | 9, 10 | imbi12d 344 | . . . . . . . . . . 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 481 | . . . . . . . 8 β’ ((π₯ β π΄ β§ π¦ β π΄) β π¦ β β) |
15 | 7, 14 | jca 511 | . . . . . . 7 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ β β β§ π¦ β β)) |
16 | ovmpot 7565 | . . . . . . 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 2732 | . . . . 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 2812 | . . . 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 3191 | . 2 β’ βπ₯ β π΄ βπ¦ β π΄ (π₯(π’ β β, π£ β β β¦ (π’ Β· π£))π¦) β π΄ |
22 | cnring 21279 | . . 3 β’ βfld β Ring | |
23 | cnfldbas 21244 | . . . 4 β’ β = (Baseββfld) | |
24 | cnfld1 21282 | . . . 4 β’ 1 = (1rββfld) | |
25 | mpocnfldmul 21247 | . . . 4 β’ (π’ β β, π£ β β β¦ (π’ Β· π£)) = (.rββfld) | |
26 | 23, 24, 25 | issubrg2 20494 | . . 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 395 β§ w3a 1084 βwal 1531 = wceq 1533 β wcel 2098 βwral 3055 βcfv 6537 (class class class)co 7405 β cmpo 7407 βcc 11110 1c1 11113 + caddc 11115 Β· cmul 11117 -cneg 11449 SubGrpcsubg 19047 Ringcrg 20138 SubRingcsubrg 20469 βfldccnfld 21240 |
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-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 ax-addf 11191 |
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-tp 4628 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-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 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-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-subrng 20446 df-subrg 20471 df-cnfld 21241 |
This theorem is referenced by: cnsubdrglem 21312 zsubrg 21314 gzsubrg 21315 cnstrcvs 25023 cncvs 25027 |
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