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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 3288 | . . . . 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 2736 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 3 | eqid 2736 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 4 | opprirred.1 | . . . . . . . 8 ⊢ 𝑆 = (oppr‘𝑅) | |
| 5 | eqid 2736 | . . . . . . . 8 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
| 6 | 2, 3, 4, 5 | opprmul 20338 | . . . . . . 7 ⊢ (𝑦(.r‘𝑆)𝑧) = (𝑧(.r‘𝑅)𝑦) | 
| 7 | 6 | neeq1i 3004 | . . . . . 6 ⊢ ((𝑦(.r‘𝑆)𝑧) ≠ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ≠ 𝑥) | 
| 8 | 7 | 2ralbii 3127 | . . . . 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 278 | . . . 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 623 | . . 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 2736 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
| 12 | opprirred.2 | . . . 4 ⊢ 𝐼 = (Irred‘𝑅) | |
| 13 | eqid 2736 | . . . 4 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
| 14 | 2, 11, 12, 13, 3 | isirred 20420 | . . 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 20342 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑆) | 
| 16 | 11, 4 | opprunit 20378 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑆) | 
| 17 | eqid 2736 | . . . 4 ⊢ (Irred‘𝑆) = (Irred‘𝑆) | |
| 18 | 15, 16, 17, 13, 5 | isirred 20420 | . . 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 303 | . 2 ⊢ (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (Irred‘𝑆)) | 
| 20 | 19 | eqriv 2733 | 1 ⊢ 𝐼 = (Irred‘𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∖ cdif 3947 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 .rcmulr 17299 opprcoppr 20334 Unitcui 20356 Irredcir 20357 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-0g 17487 df-mgp 20139 df-ur 20180 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-irred 20360 | 
| This theorem is referenced by: irredlmul 20429 | 
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