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Mirrors > Home > MPE Home > Th. List > opprnzr | Structured version Visualization version GIF version |
Description: The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.) |
Ref | Expression |
---|---|
opprnzr.1 | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprnzr | ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20654 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | opprnzr.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
3 | 2 | opprring 19983 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ Ring) |
5 | eqid 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 5 | isnzr2 20656 | . . 3 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ (Base‘𝑅))) |
7 | 6 | simprbi 497 | . 2 ⊢ (𝑅 ∈ NzRing → 2o ≼ (Base‘𝑅)) |
8 | 2, 5 | opprbas 19979 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑂) |
9 | 8 | isnzr2 20656 | . 2 ⊢ (𝑂 ∈ NzRing ↔ (𝑂 ∈ Ring ∧ 2o ≼ (Base‘𝑅))) |
10 | 4, 7, 9 | sylanbrc 583 | 1 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6491 2oc2o 8373 ≼ cdom 8814 Basecbs 17017 Ringcrg 19888 opprcoppr 19971 NzRingcnzr 20650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-tpos 8124 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-plusg 17080 df-mulr 17081 df-0g 17257 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-grp 18685 df-minusg 18686 df-mgp 19826 df-ur 19843 df-ring 19890 df-oppr 19972 df-nzr 20651 |
This theorem is referenced by: opprdomn 20694 |
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