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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 21205 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | opprdomn.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | 2 | opprnzr 20422 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ NzRing) |
5 | eqid 2726 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2726 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | domneq0 21207 | . . . . . . 7 ⊢ ((𝑅 ∈ Domn ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
9 | 8 | 3com23 1123 | . . . . . 6 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
10 | eqid 2726 | . . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
11 | 5, 6, 2, 10 | opprmul 20239 | . . . . . . 7 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
12 | 11 | eqeq1i 2731 | . . . . . 6 ⊢ ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑅)𝑥) = (0g‘𝑅)) |
13 | orcom 867 | . . . . . 6 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅))) | |
14 | 9, 12, 13 | 3bitr4g 314 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
15 | 14 | biimpd 228 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
16 | 15 | 3expb 1117 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
17 | 16 | ralrimivva 3194 | . 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 20243 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
19 | 2, 7 | oppr0 20251 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
20 | 18, 10, 19 | isdomn 21204 | . 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 582 | 1 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 .rcmulr 17207 0gc0g 17394 opprcoppr 20235 NzRingcnzr 20414 Domncdomn 21190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-nzr 20415 df-domn 21194 |
This theorem is referenced by: fidomndrng 21222 |
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