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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 3287 | . . . . 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 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | eqid 2735 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | opprirred.1 | . . . . . . . 8 ⊢ 𝑆 = (oppr‘𝑅) | |
5 | eqid 2735 | . . . . . . . 8 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
6 | 2, 3, 4, 5 | opprmul 20354 | . . . . . . 7 ⊢ (𝑦(.r‘𝑆)𝑧) = (𝑧(.r‘𝑅)𝑦) |
7 | 6 | neeq1i 3003 | . . . . . 6 ⊢ ((𝑦(.r‘𝑆)𝑧) ≠ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ≠ 𝑥) |
8 | 7 | 2ralbii 3126 | . . . . 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 2735 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
12 | opprirred.2 | . . . 4 ⊢ 𝐼 = (Irred‘𝑅) | |
13 | eqid 2735 | . . . 4 ⊢ ((Base‘𝑅) ∖ (Unit‘𝑅)) = ((Base‘𝑅) ∖ (Unit‘𝑅)) | |
14 | 2, 11, 12, 13, 3 | isirred 20436 | . . 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 20358 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑆) |
16 | 11, 4 | opprunit 20394 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑆) |
17 | eqid 2735 | . . . 4 ⊢ (Irred‘𝑆) = (Irred‘𝑆) | |
18 | 15, 16, 17, 13, 5 | isirred 20436 | . . 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 2732 | 1 ⊢ 𝐼 = (Irred‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∖ cdif 3960 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 opprcoppr 20350 Unitcui 20372 Irredcir 20373 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgp 20153 df-ur 20200 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-irred 20376 |
This theorem is referenced by: irredlmul 20445 |
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