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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprqus1r | Structured version Visualization version GIF version |
Description: The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
opprqus.b | β’ π΅ = (Baseβπ ) |
opprqus.o | β’ π = (opprβπ ) |
opprqus.q | β’ π = (π /s (π ~QG πΌ)) |
opprqus1r.r | β’ (π β π β Ring) |
opprqus1r.i | β’ (π β πΌ β (2Idealβπ )) |
Ref | Expression |
---|---|
opprqus1r | β’ (π β (1rβ(opprβπ)) = (1rβ(π /s (π ~QG πΌ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . 2 β’ (Baseβ(opprβπ)) = (Baseβ(opprβπ)) | |
2 | fvexd 6896 | . 2 β’ (π β (opprβπ) β V) | |
3 | ovexd 7436 | . 2 β’ (π β (π /s (π ~QG πΌ)) β V) | |
4 | opprqus.b | . . 3 β’ π΅ = (Baseβπ ) | |
5 | opprqus.o | . . 3 β’ π = (opprβπ ) | |
6 | opprqus.q | . . 3 β’ π = (π /s (π ~QG πΌ)) | |
7 | opprqus1r.r | . . 3 β’ (π β π β Ring) | |
8 | opprqus1r.i | . . . . 5 β’ (π β πΌ β (2Idealβπ )) | |
9 | 8 | 2idllidld 21100 | . . . 4 β’ (π β πΌ β (LIdealβπ )) |
10 | eqid 2724 | . . . . 5 β’ (LIdealβπ ) = (LIdealβπ ) | |
11 | 4, 10 | lidlss 21060 | . . . 4 β’ (πΌ β (LIdealβπ ) β πΌ β π΅) |
12 | 9, 11 | syl 17 | . . 3 β’ (π β πΌ β π΅) |
13 | 4, 5, 6, 7, 12 | opprqusbas 33037 | . 2 β’ (π β (Baseβ(opprβπ)) = (Baseβ(π /s (π ~QG πΌ)))) |
14 | 7 | ad2antrr 723 | . . 3 β’ (((π β§ π₯ β (Baseβ(opprβπ))) β§ π¦ β (Baseβ(opprβπ))) β π β Ring) |
15 | 8 | ad2antrr 723 | . . 3 β’ (((π β§ π₯ β (Baseβ(opprβπ))) β§ π¦ β (Baseβ(opprβπ))) β πΌ β (2Idealβπ )) |
16 | eqid 2724 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
17 | simpr 484 | . . . . 5 β’ ((π β§ π₯ β (Baseβ(opprβπ))) β π₯ β (Baseβ(opprβπ))) | |
18 | eqid 2724 | . . . . . 6 β’ (opprβπ) = (opprβπ) | |
19 | 18, 16 | opprbas 20232 | . . . . 5 β’ (Baseβπ) = (Baseβ(opprβπ)) |
20 | 17, 19 | eleqtrrdi 2836 | . . . 4 β’ ((π β§ π₯ β (Baseβ(opprβπ))) β π₯ β (Baseβπ)) |
21 | 20 | adantr 480 | . . 3 β’ (((π β§ π₯ β (Baseβ(opprβπ))) β§ π¦ β (Baseβ(opprβπ))) β π₯ β (Baseβπ)) |
22 | simpr 484 | . . . . 5 β’ ((π β§ π¦ β (Baseβ(opprβπ))) β π¦ β (Baseβ(opprβπ))) | |
23 | 22, 19 | eleqtrrdi 2836 | . . . 4 β’ ((π β§ π¦ β (Baseβ(opprβπ))) β π¦ β (Baseβπ)) |
24 | 23 | adantlr 712 | . . 3 β’ (((π β§ π₯ β (Baseβ(opprβπ))) β§ π¦ β (Baseβ(opprβπ))) β π¦ β (Baseβπ)) |
25 | 4, 5, 6, 14, 15, 16, 21, 24 | opprqusmulr 33040 | . 2 β’ (((π β§ π₯ β (Baseβ(opprβπ))) β§ π¦ β (Baseβ(opprβπ))) β (π₯(.rβ(opprβπ))π¦) = (π₯(.rβ(π /s (π ~QG πΌ)))π¦)) |
26 | 1, 2, 3, 13, 25 | urpropd 32812 | 1 β’ (π β (1rβ(opprβπ)) = (1rβ(π /s (π ~QG πΌ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3466 β wss 3940 βcfv 6533 (class class class)co 7401 Basecbs 17142 /s cqus 17449 ~QG cqg 19038 1rcur 20075 Ringcrg 20127 opprcoppr 20224 LIdealclidl 21054 2Idealc2idl 21095 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-ec 8700 df-qs 8704 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-0g 17385 df-imas 17452 df-qus 17453 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-eqg 19041 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-subrg 20460 df-lmod 20697 df-lss 20768 df-sra 21010 df-rgmod 21011 df-lidl 21056 df-2idl 21096 |
This theorem is referenced by: opprqusdrng 33042 qsdrngi 33044 |
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