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Mirrors > Home > MPE Home > Th. List > opprrngb | Structured version Visualization version GIF version |
Description: A class is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 20298. (Contributed by AV, 15-Feb-2025.) |
Ref | Expression |
---|---|
opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprrngb | ⊢ (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprbas.1 | . . 3 ⊢ 𝑂 = (oppr‘𝑅) | |
2 | 1 | opprrng 20298 | . 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng) |
3 | eqid 2728 | . . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂) | |
4 | 3 | opprrng 20298 | . . 3 ⊢ (𝑂 ∈ Rng → (oppr‘𝑂) ∈ Rng) |
5 | eqidd 2729 | . . . . 5 ⊢ (⊤ → (Base‘𝑅) = (Base‘𝑅)) | |
6 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 1, 6 | opprbas 20294 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑂) |
8 | 3, 7 | opprbas 20294 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑂)) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (⊤ → (Base‘𝑅) = (Base‘(oppr‘𝑂))) |
10 | eqid 2728 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
11 | 1, 10 | oppradd 20296 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑂) |
12 | 3, 11 | oppradd 20296 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑂)) |
13 | 12 | oveqi 7439 | . . . . . 6 ⊢ (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦) |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦)) |
15 | eqid 2728 | . . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
16 | eqid 2728 | . . . . . . . 8 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂)) | |
17 | 7, 15, 3, 16 | opprmul 20290 | . . . . . . 7 ⊢ (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥) |
18 | eqid 2728 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 6, 18, 1, 15 | opprmul 20290 | . . . . . . 7 ⊢ (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦) |
20 | 17, 19 | eqtr2i 2757 | . . . . . 6 ⊢ (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦) |
21 | 20 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦)) |
22 | 5, 9, 14, 21 | rngpropd 20128 | . . . 4 ⊢ (⊤ → (𝑅 ∈ Rng ↔ (oppr‘𝑂) ∈ Rng)) |
23 | 22 | mptru 1540 | . . 3 ⊢ (𝑅 ∈ Rng ↔ (oppr‘𝑂) ∈ Rng) |
24 | 4, 23 | sylibr 233 | . 2 ⊢ (𝑂 ∈ Rng → 𝑅 ∈ Rng) |
25 | 2, 24 | impbii 208 | 1 ⊢ (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 .rcmulr 17243 Rngcrng 20106 opprcoppr 20286 |
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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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-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 7879 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-mulr 17256 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-oppr 20287 |
This theorem is referenced by: opprsubrng 20510 |
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