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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 3307 | . . . . 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 2798 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | opprirred.1 | . . . . . . . 8 ⊢ 𝑆 = (oppr‘𝑅) | |
5 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
6 | 2, 3, 4, 5 | opprmul 19372 | . . . . . . 7 ⊢ (𝑦(.r‘𝑆)𝑧) = (𝑧(.r‘𝑅)𝑦) |
7 | 6 | neeq1i 3051 | . . . . . 6 ⊢ ((𝑦(.r‘𝑆)𝑧) ≠ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ≠ 𝑥) |
8 | 7 | 2ralbii 3134 | . . . . 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 281 | . . . 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 625 | . . 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 2798 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
12 | opprirred.2 | . . . 4 ⊢ 𝐼 = (Irred‘𝑅) | |
13 | eqid 2798 | . . . 4 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
14 | 2, 11, 12, 13, 3 | isirred 19445 | . . 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 19375 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑆) |
16 | 11, 4 | opprunit 19407 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑆) |
17 | eqid 2798 | . . . 4 ⊢ (Irred‘𝑆) = (Irred‘𝑆) | |
18 | 15, 16, 17, 13, 5 | isirred 19445 | . . 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 306 | . 2 ⊢ (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (Irred‘𝑆)) |
20 | 19 | eqriv 2795 | 1 ⊢ 𝐼 = (Irred‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∖ cdif 3878 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 opprcoppr 19368 Unitcui 19385 Irredcir 19386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgp 19233 df-ur 19245 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-irred 19389 |
This theorem is referenced by: irredlmul 19454 |
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