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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 3276 | . . . . 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 2725 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2725 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | opprirred.1 | . . . . . . . 8 ⊢ 𝑆 = (oppr‘𝑅) | |
5 | eqid 2725 | . . . . . . . 8 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
6 | 2, 3, 4, 5 | opprmul 20288 | . . . . . . 7 ⊢ (𝑦(.r‘𝑆)𝑧) = (𝑧(.r‘𝑅)𝑦) |
7 | 6 | neeq1i 2994 | . . . . . 6 ⊢ ((𝑦(.r‘𝑆)𝑧) ≠ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ≠ 𝑥) |
8 | 7 | 2ralbii 3117 | . . . . 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 277 | . . . 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 621 | . . 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 2725 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
12 | opprirred.2 | . . . 4 ⊢ 𝐼 = (Irred‘𝑅) | |
13 | eqid 2725 | . . . 4 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
14 | 2, 11, 12, 13, 3 | isirred 20370 | . . 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 20292 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑆) |
16 | 11, 4 | opprunit 20328 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑆) |
17 | eqid 2725 | . . . 4 ⊢ (Irred‘𝑆) = (Irred‘𝑆) | |
18 | 15, 16, 17, 13, 5 | isirred 20370 | . . 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 302 | . 2 ⊢ (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (Irred‘𝑆)) |
20 | 19 | eqriv 2722 | 1 ⊢ 𝐼 = (Irred‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∖ cdif 3941 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 .rcmulr 17237 opprcoppr 20284 Unitcui 20306 Irredcir 20307 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-0g 17426 df-mgp 20087 df-ur 20134 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-irred 20310 |
This theorem is referenced by: irredlmul 20379 |
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