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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 | ⊢ 𝑆 = (oppr‘𝑅) |
opprirred.2 | ⊢ 𝐼 = (Irred‘𝑅) |
Ref | Expression |
---|---|
opprirred | ⊢ 𝐼 = (Irred‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3354 | . . . . 5 ⊢ (∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑧(.r‘𝑅)𝑦) ≠ 𝑥 ↔ ∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑧(.r‘𝑅)𝑦) ≠ 𝑥) | |
2 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | opprirred.1 | . . . . . . . 8 ⊢ 𝑆 = (oppr‘𝑅) | |
5 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
6 | 2, 3, 4, 5 | opprmul 19370 | . . . . . . 7 ⊢ (𝑦(.r‘𝑆)𝑧) = (𝑧(.r‘𝑅)𝑦) |
7 | 6 | neeq1i 3080 | . . . . . 6 ⊢ ((𝑦(.r‘𝑆)𝑧) ≠ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ≠ 𝑥) |
8 | 7 | 2ralbii 3166 | . . . . 5 ⊢ (∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑦(.r‘𝑆)𝑧) ≠ 𝑥 ↔ ∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑧(.r‘𝑅)𝑦) ≠ 𝑥) |
9 | 1, 8 | bitr4i 280 | . . . 4 ⊢ (∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑧(.r‘𝑅)𝑦) ≠ 𝑥 ↔ ∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑦(.r‘𝑆)𝑧) ≠ 𝑥) |
10 | 9 | anbi2i 624 | . . 3 ⊢ ((𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ∧ ∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑧(.r‘𝑅)𝑦) ≠ 𝑥) ↔ (𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ∧ ∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑦(.r‘𝑆)𝑧) ≠ 𝑥)) |
11 | eqid 2821 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
12 | opprirred.2 | . . . 4 ⊢ 𝐼 = (Irred‘𝑅) | |
13 | eqid 2821 | . . . 4 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
14 | 2, 11, 12, 13, 3 | isirred 19443 | . . 3 ⊢ (𝑥 ∈ 𝐼 ↔ (𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ∧ ∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑧(.r‘𝑅)𝑦) ≠ 𝑥)) |
15 | 4, 2 | opprbas 19373 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑆) |
16 | 11, 4 | opprunit 19405 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑆) |
17 | eqid 2821 | . . . 4 ⊢ (Irred‘𝑆) = (Irred‘𝑆) | |
18 | 15, 16, 17, 13, 5 | isirred 19443 | . . 3 ⊢ (𝑥 ∈ (Irred‘𝑆) ↔ (𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)) ∧ ∀𝑦 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))∀𝑧 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))(𝑦(.r‘𝑆)𝑧) ≠ 𝑥)) |
19 | 10, 14, 18 | 3bitr4i 305 | . 2 ⊢ (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (Irred‘𝑆)) |
20 | 19 | eqriv 2818 | 1 ⊢ 𝐼 = (Irred‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∖ cdif 3933 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 .rcmulr 16560 opprcoppr 19366 Unitcui 19383 Irredcir 19384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgp 19234 df-ur 19246 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-irred 19387 |
This theorem is referenced by: irredlmul 19452 |
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