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Mirrors > Home > MPE Home > Th. List > rngqiprngghmlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rngqiprngghm 21178. (Contributed by AV, 25-Feb-2025.) |
Ref | Expression |
---|---|
rng2idlring.r | β’ (π β π β Rng) |
rng2idlring.i | β’ (π β πΌ β (2Idealβπ )) |
rng2idlring.j | β’ π½ = (π βΎs πΌ) |
rng2idlring.u | β’ (π β π½ β Ring) |
rng2idlring.b | β’ π΅ = (Baseβπ ) |
rng2idlring.t | β’ Β· = (.rβπ ) |
rng2idlring.1 | β’ 1 = (1rβπ½) |
Ref | Expression |
---|---|
rngqiprngghmlem1 | β’ ((π β§ π΄ β π΅) β ( 1 Β· π΄) β (Baseβπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rng2idlring.r | . . 3 β’ (π β π β Rng) | |
2 | rng2idlring.i | . . . . 5 β’ (π β πΌ β (2Idealβπ )) | |
3 | rng2idlring.j | . . . . 5 β’ π½ = (π βΎs πΌ) | |
4 | eqid 2727 | . . . . 5 β’ (Baseβπ½) = (Baseβπ½) | |
5 | 2, 3, 4 | 2idlelbas 21147 | . . . 4 β’ (π β ((Baseβπ½) β (LIdealβπ ) β§ (Baseβπ½) β (LIdealβ(opprβπ )))) |
6 | 5 | simprd 495 | . . 3 β’ (π β (Baseβπ½) β (LIdealβ(opprβπ ))) |
7 | rng2idlring.u | . . . . . . 7 β’ (π β π½ β Ring) | |
8 | ringrng 20210 | . . . . . . 7 β’ (π½ β Ring β π½ β Rng) | |
9 | 7, 8 | syl 17 | . . . . . 6 β’ (π β π½ β Rng) |
10 | 3, 9 | eqeltrrid 2833 | . . . . 5 β’ (π β (π βΎs πΌ) β Rng) |
11 | 1, 2, 10 | rng2idl0 21150 | . . . 4 β’ (π β (0gβπ ) β πΌ) |
12 | 2, 3, 4 | 2idlbas 21146 | . . . 4 β’ (π β (Baseβπ½) = πΌ) |
13 | 11, 12 | eleqtrrd 2831 | . . 3 β’ (π β (0gβπ ) β (Baseβπ½)) |
14 | 1, 6, 13 | 3jca 1126 | . 2 β’ (π β (π β Rng β§ (Baseβπ½) β (LIdealβ(opprβπ )) β§ (0gβπ ) β (Baseβπ½))) |
15 | rng2idlring.1 | . . . . 5 β’ 1 = (1rβπ½) | |
16 | 4, 15 | ringidcl 20191 | . . . 4 β’ (π½ β Ring β 1 β (Baseβπ½)) |
17 | 7, 16 | syl 17 | . . 3 β’ (π β 1 β (Baseβπ½)) |
18 | 17 | anim1ci 615 | . 2 β’ ((π β§ π΄ β π΅) β (π΄ β π΅ β§ 1 β (Baseβπ½))) |
19 | eqid 2727 | . . 3 β’ (0gβπ ) = (0gβπ ) | |
20 | rng2idlring.b | . . 3 β’ π΅ = (Baseβπ ) | |
21 | rng2idlring.t | . . 3 β’ Β· = (.rβπ ) | |
22 | eqid 2727 | . . 3 β’ (LIdealβ(opprβπ )) = (LIdealβ(opprβπ )) | |
23 | 19, 20, 21, 22 | rngridlmcl 21102 | . 2 β’ (((π β Rng β§ (Baseβπ½) β (LIdealβ(opprβπ )) β§ (0gβπ ) β (Baseβπ½)) β§ (π΄ β π΅ β§ 1 β (Baseβπ½))) β ( 1 Β· π΄) β (Baseβπ½)) |
24 | 14, 18, 23 | syl2an2r 684 | 1 β’ ((π β§ π΄ β π΅) β ( 1 Β· π΄) β (Baseβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 Basecbs 17171 βΎs cress 17200 .rcmulr 17225 0gc0g 17412 Rngcrng 20083 1rcur 20112 Ringcrg 20164 opprcoppr 20261 LIdealclidl 21091 2Idealc2idl 21132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-subg 19069 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-subrng 20472 df-lss 20805 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-2idl 21133 |
This theorem is referenced by: rngqiprngghmlem2 21167 rngqiprngimfolem 21169 rngqiprnglinlem1 21170 rngqiprngghm 21178 rngqiprngimfo 21180 rngqiprnglin 21181 rng2idl1cntr 21184 rngqiprngfulem4 21193 |
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