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Mirrors > Home > MPE Home > Th. List > opprsubrg | Structured version Visualization version GIF version |
Description: Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
Ref | Expression |
---|---|
opprsubrg.o | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprsubrg | ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgrcl 19662 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | subrgrcl 19662 | . . . 4 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑂 ∈ Ring) | |
3 | opprsubrg.o | . . . . 5 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | 3 | opprringb 19507 | . . . 4 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring) |
5 | 2, 4 | sylibr 237 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑅 ∈ Ring) |
6 | 3 | opprsubg 19511 | . . . . . . 7 ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘𝑂)) |
8 | 7 | eleq2d 2819 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂))) |
9 | ralcom 3259 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) | |
10 | eqid 2739 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2739 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | eqid 2739 | . . . . . . . . . 10 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
13 | 10, 11, 3, 12 | opprmul 19501 | . . . . . . . . 9 ⊢ (𝑧(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑧) |
14 | 13 | eleq1i 2824 | . . . . . . . 8 ⊢ ((𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
15 | 14 | 2ralbii 3082 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
16 | 9, 15 | bitr4i 281 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥) |
17 | 16 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥)) |
18 | 8, 17 | 3anbi13d 1439 | . . . 4 ⊢ (𝑅 ∈ Ring → ((𝑥 ∈ (SubGrp‘𝑅) ∧ (1r‘𝑅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) ↔ (𝑥 ∈ (SubGrp‘𝑂) ∧ (1r‘𝑅) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥))) |
19 | eqid 2739 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 10, 19, 11 | issubrg2 19677 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ (1r‘𝑅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥))) |
21 | 3, 10 | opprbas 19504 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂) |
22 | 3, 19 | oppr1 19509 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑂) |
23 | 21, 22, 12 | issubrg2 19677 | . . . . 5 ⊢ (𝑂 ∈ Ring → (𝑥 ∈ (SubRing‘𝑂) ↔ (𝑥 ∈ (SubGrp‘𝑂) ∧ (1r‘𝑅) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥))) |
24 | 4, 23 | sylbi 220 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑂) ↔ (𝑥 ∈ (SubGrp‘𝑂) ∧ (1r‘𝑅) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥))) |
25 | 18, 20, 24 | 3bitr4d 314 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂))) |
26 | 1, 5, 25 | pm5.21nii 383 | . 2 ⊢ (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂)) |
27 | 26 | eqriv 2736 | 1 ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 .rcmulr 16672 SubGrpcsubg 18394 1rcur 19373 Ringcrg 19419 opprcoppr 19497 SubRingcsubrg 19653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-tpos 7924 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-0g 16821 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-grp 18225 df-subg 18397 df-mgp 19362 df-ur 19374 df-ring 19421 df-oppr 19498 df-subrg 19655 |
This theorem is referenced by: (None) |
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