Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19295 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | opprdomn.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | 2 | opprnzr 19265 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ NzRing) |
5 | eqid 2622 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2622 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | eqid 2622 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
8 | 5, 6, 7 | domneq0 19297 | . . . . . . 7 ⊢ ((𝑅 ∈ Domn ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
9 | 8 | 3com23 1271 | . . . . . 6 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅)))) |
10 | eqid 2622 | . . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
11 | 5, 6, 2, 10 | opprmul 18626 | . . . . . . 7 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
12 | 11 | eqeq1i 2627 | . . . . . 6 ⊢ ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑅)𝑥) = (0g‘𝑅)) |
13 | orcom 402 | . . . . . 6 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑅) ∨ 𝑥 = (0g‘𝑅))) | |
14 | 9, 12, 13 | 3bitr4g 303 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) ↔ (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
15 | 14 | biimpd 219 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
16 | 15 | 3expb 1266 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) |
17 | 16 | ralrimivva 2971 | . 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 18629 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
19 | 2, 7 | oppr0 18633 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑂) |
20 | 18, 10, 19 | isdomn 19294 | . 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 698 | 1 ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 0gc0g 16100 opprcoppr 18622 NzRingcnzr 19257 Domncdomn 19280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-nzr 19258 df-domn 19284 |
This theorem is referenced by: fidomndrng 19307 |
Copyright terms: Public domain | W3C validator |