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Mirrors > Home > MPE Home > Th. List > rngqiprngghmlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for rngqiprngghm 21178. (Contributed by AV, 25-Feb-2025.) (Proof shortened by AV, 24-Mar-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 |
---|---|
rngqiprngghmlem3 | β’ ((π β§ (π΄ β π΅ β§ πΆ β π΅)) β ( 1 Β· (π΄(+gβπ )πΆ)) = (( 1 Β· π΄)(+gβπ½)( 1 Β· πΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rng2idlring.r | . . 3 β’ (π β π β Rng) | |
2 | rng2idlring.i | . . . . . 6 β’ (π β πΌ β (2Idealβπ )) | |
3 | rng2idlring.j | . . . . . 6 β’ π½ = (π βΎs πΌ) | |
4 | rng2idlring.u | . . . . . 6 β’ (π β π½ β Ring) | |
5 | rng2idlring.b | . . . . . 6 β’ π΅ = (Baseβπ ) | |
6 | rng2idlring.t | . . . . . 6 β’ Β· = (.rβπ ) | |
7 | rng2idlring.1 | . . . . . 6 β’ 1 = (1rβπ½) | |
8 | 1, 2, 3, 4, 5, 6, 7 | rngqiprng1elbas 21165 | . . . . 5 β’ (π β 1 β π΅) |
9 | 8 | anim1i 614 | . . . 4 β’ ((π β§ (π΄ β π΅ β§ πΆ β π΅)) β ( 1 β π΅ β§ (π΄ β π΅ β§ πΆ β π΅))) |
10 | 3anass 1093 | . . . 4 β’ (( 1 β π΅ β§ π΄ β π΅ β§ πΆ β π΅) β ( 1 β π΅ β§ (π΄ β π΅ β§ πΆ β π΅))) | |
11 | 9, 10 | sylibr 233 | . . 3 β’ ((π β§ (π΄ β π΅ β§ πΆ β π΅)) β ( 1 β π΅ β§ π΄ β π΅ β§ πΆ β π΅)) |
12 | eqid 2727 | . . . 4 β’ (+gβπ ) = (+gβπ ) | |
13 | 5, 12, 6 | rngdi 20091 | . . 3 β’ ((π β Rng β§ ( 1 β π΅ β§ π΄ β π΅ β§ πΆ β π΅)) β ( 1 Β· (π΄(+gβπ )πΆ)) = (( 1 Β· π΄)(+gβπ )( 1 Β· πΆ))) |
14 | 1, 11, 13 | syl2an2r 684 | . 2 β’ ((π β§ (π΄ β π΅ β§ πΆ β π΅)) β ( 1 Β· (π΄(+gβπ )πΆ)) = (( 1 Β· π΄)(+gβπ )( 1 Β· πΆ))) |
15 | 3, 12 | ressplusg 17262 | . . . . 5 β’ (πΌ β (2Idealβπ ) β (+gβπ ) = (+gβπ½)) |
16 | 2, 15 | syl 17 | . . . 4 β’ (π β (+gβπ ) = (+gβπ½)) |
17 | 16 | oveqd 7431 | . . 3 β’ (π β (( 1 Β· π΄)(+gβπ )( 1 Β· πΆ)) = (( 1 Β· π΄)(+gβπ½)( 1 Β· πΆ))) |
18 | 17 | adantr 480 | . 2 β’ ((π β§ (π΄ β π΅ β§ πΆ β π΅)) β (( 1 Β· π΄)(+gβπ )( 1 Β· πΆ)) = (( 1 Β· π΄)(+gβπ½)( 1 Β· πΆ))) |
19 | 14, 18 | eqtrd 2767 | 1 β’ ((π β§ (π΄ β π΅ β§ πΆ β π΅)) β ( 1 Β· (π΄(+gβπ )πΆ)) = (( 1 Β· π΄)(+gβπ½)( 1 Β· πΆ))) |
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 +gcplusg 17224 .rcmulr 17225 Rngcrng 20083 1rcur 20112 Ringcrg 20164 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-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-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-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 |
This theorem is referenced by: rngqiprngghm 21178 |
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