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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 19116 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | opprdomn.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | 2 | opprnzr 19086 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ NzRing) |
5 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | eqid 2610 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | domneq0 19118 | . . . . . . 7 ⊢ ((𝑅 ∈ Domn ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
9 | 8 | 3com23 1263 | . . . . . 6 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
10 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
11 | 5, 6, 2, 10 | opprmul 18449 | . . . . . . 7 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
12 | 11 | eqeq1i 2615 | . . . . . 6 ⊢ ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑅)𝑥) = (0g‘𝑅)) |
13 | orcom 401 | . . . . . 6 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅))) | |
14 | 9, 12, 13 | 3bitr4g 302 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
15 | 14 | biimpd 218 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
16 | 15 | 3expb 1258 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
17 | 16 | ralrimivva 2954 | . 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 18452 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
19 | 2, 7 | oppr0 18456 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
20 | 18, 10, 19 | isdomn 19115 | . 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 695 | 1 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 .rcmulr 15769 0gc0g 15923 opprcoppr 18445 NzRingcnzr 19078 Domncdomn 19101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-nzr 19079 df-domn 19105 |
This theorem is referenced by: fidomndrng 19128 |
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