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Mirrors > Home > MPE Home > Th. List > opprdomn | Structured version Visualization version GIF version |
Description: The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
opprdomn.1 | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprdomn | ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnnzr 20070 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | opprdomn.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | 2 | opprnzr 20040 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ NzRing) |
5 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2823 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | eqid 2823 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | domneq0 20072 | . . . . . . 7 ⊢ ((𝑅 ∈ Domn ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
9 | 8 | 3com23 1122 | . . . . . 6 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
10 | eqid 2823 | . . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
11 | 5, 6, 2, 10 | opprmul 19378 | . . . . . . 7 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
12 | 11 | eqeq1i 2828 | . . . . . 6 ⊢ ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑅)𝑥) = (0g‘𝑅)) |
13 | orcom 866 | . . . . . 6 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅))) | |
14 | 9, 12, 13 | 3bitr4g 316 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
15 | 14 | biimpd 231 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
16 | 15 | 3expb 1116 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
17 | 16 | ralrimivva 3193 | . 2 ⊢ (𝑅 ∈ Domn → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
18 | 2, 5 | opprbas 19381 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
19 | 2, 7 | oppr0 19385 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
20 | 18, 10, 19 | isdomn 20069 | . 2 ⊢ (𝑂 ∈ Domn ↔ (𝑂 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))))) |
21 | 4, 17, 20 | sylanbrc 585 | 1 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 0gc0g 16715 opprcoppr 19374 NzRingcnzr 20032 Domncdomn 20055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-nzr 20033 df-domn 20059 |
This theorem is referenced by: fidomndrng 20082 |
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