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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 21249 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | opprdomn.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | 2 | opprnzr 20466 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ NzRing) |
5 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2728 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | eqid 2728 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | domneq0 21251 | . . . . . . 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 2728 | . . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
11 | 5, 6, 2, 10 | opprmul 20283 | . . . . . . 7 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
12 | 11 | eqeq1i 2733 | . . . . . 6 ⊢ ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑅)𝑥) = (0g‘𝑅)) |
13 | orcom 868 | . . . . . 6 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅))) | |
14 | 9, 12, 13 | 3bitr4g 313 | . . . . 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 3198 | . 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 20287 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
19 | 2, 7 | oppr0 20295 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
20 | 18, 10, 19 | isdomn 21248 | . 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 581 | 1 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 .rcmulr 17241 0gc0g 17428 opprcoppr 20279 NzRingcnzr 20458 Domncdomn 21234 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-nzr 20459 df-domn 21238 |
This theorem is referenced by: fidomndrng 21268 |
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