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Mirrors > Home > MPE Home > Th. List > opprirred | Structured version Visualization version GIF version |
Description: Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
opprirred.1 | β’ π = (opprβπ ) |
opprirred.2 | β’ πΌ = (Irredβπ ) |
Ref | Expression |
---|---|
opprirred | β’ πΌ = (Irredβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3273 | . . . . 5 β’ (βπ§ β ((Baseβπ ) β (Unitβπ ))βπ¦ β ((Baseβπ ) β (Unitβπ ))(π§(.rβπ )π¦) β π₯ β βπ¦ β ((Baseβπ ) β (Unitβπ ))βπ§ β ((Baseβπ ) β (Unitβπ ))(π§(.rβπ )π¦) β π₯) | |
2 | eqid 2737 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2737 | . . . . . . . 8 β’ (.rβπ ) = (.rβπ ) | |
4 | opprirred.1 | . . . . . . . 8 β’ π = (opprβπ ) | |
5 | eqid 2737 | . . . . . . . 8 β’ (.rβπ) = (.rβπ) | |
6 | 2, 3, 4, 5 | opprmul 20053 | . . . . . . 7 β’ (π¦(.rβπ)π§) = (π§(.rβπ )π¦) |
7 | 6 | neeq1i 3009 | . . . . . 6 β’ ((π¦(.rβπ)π§) β π₯ β (π§(.rβπ )π¦) β π₯) |
8 | 7 | 2ralbii 3128 | . . . . 5 β’ (βπ¦ β ((Baseβπ ) β (Unitβπ ))βπ§ β ((Baseβπ ) β (Unitβπ ))(π¦(.rβπ)π§) β π₯ β βπ¦ β ((Baseβπ ) β (Unitβπ ))βπ§ β ((Baseβπ ) β (Unitβπ ))(π§(.rβπ )π¦) β π₯) |
9 | 1, 8 | bitr4i 278 | . . . 4 β’ (βπ§ β ((Baseβπ ) β (Unitβπ ))βπ¦ β ((Baseβπ ) β (Unitβπ ))(π§(.rβπ )π¦) β π₯ β βπ¦ β ((Baseβπ ) β (Unitβπ ))βπ§ β ((Baseβπ ) β (Unitβπ ))(π¦(.rβπ)π§) β π₯) |
10 | 9 | anbi2i 624 | . . 3 β’ ((π₯ β ((Baseβπ ) β (Unitβπ )) β§ βπ§ β ((Baseβπ ) β (Unitβπ ))βπ¦ β ((Baseβπ ) β (Unitβπ ))(π§(.rβπ )π¦) β π₯) β (π₯ β ((Baseβπ ) β (Unitβπ )) β§ βπ¦ β ((Baseβπ ) β (Unitβπ ))βπ§ β ((Baseβπ ) β (Unitβπ ))(π¦(.rβπ)π§) β π₯)) |
11 | eqid 2737 | . . . 4 β’ (Unitβπ ) = (Unitβπ ) | |
12 | opprirred.2 | . . . 4 β’ πΌ = (Irredβπ ) | |
13 | eqid 2737 | . . . 4 β’ ((Baseβπ ) β (Unitβπ )) = ((Baseβπ ) β (Unitβπ )) | |
14 | 2, 11, 12, 13, 3 | isirred 20129 | . . 3 β’ (π₯ β πΌ β (π₯ β ((Baseβπ ) β (Unitβπ )) β§ βπ§ β ((Baseβπ ) β (Unitβπ ))βπ¦ β ((Baseβπ ) β (Unitβπ ))(π§(.rβπ )π¦) β π₯)) |
15 | 4, 2 | opprbas 20057 | . . . 4 β’ (Baseβπ ) = (Baseβπ) |
16 | 11, 4 | opprunit 20091 | . . . 4 β’ (Unitβπ ) = (Unitβπ) |
17 | eqid 2737 | . . . 4 β’ (Irredβπ) = (Irredβπ) | |
18 | 15, 16, 17, 13, 5 | isirred 20129 | . . 3 β’ (π₯ β (Irredβπ) β (π₯ β ((Baseβπ ) β (Unitβπ )) β§ βπ¦ β ((Baseβπ ) β (Unitβπ ))βπ§ β ((Baseβπ ) β (Unitβπ ))(π¦(.rβπ)π§) β π₯)) |
19 | 10, 14, 18 | 3bitr4i 303 | . 2 β’ (π₯ β πΌ β π₯ β (Irredβπ)) |
20 | 19 | eqriv 2734 | 1 β’ πΌ = (Irredβπ) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 β cdif 3908 βcfv 6497 (class class class)co 7358 Basecbs 17084 .rcmulr 17135 opprcoppr 20049 Unitcui 20069 Irredcir 20070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-0g 17324 df-mgp 19898 df-ur 19915 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-irred 20073 |
This theorem is referenced by: irredlmul 20138 |
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