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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 19469 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | subrgrcl 19469 | . . . 4 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑂 ∈ Ring) | |
3 | opprsubrg.o | . . . . 5 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | 3 | opprringb 19311 | . . . 4 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring) |
5 | 2, 4 | sylibr 235 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑅 ∈ Ring) |
6 | 3 | opprsubg 19315 | . . . . . . 7 ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘𝑂)) |
8 | 7 | eleq2d 2895 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂))) |
9 | ralcom 3351 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) | |
10 | eqid 2818 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2818 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | eqid 2818 | . . . . . . . . . 10 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
13 | 10, 11, 3, 12 | opprmul 19305 | . . . . . . . . 9 ⊢ (𝑧(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑧) |
14 | 13 | eleq1i 2900 | . . . . . . . 8 ⊢ ((𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
15 | 14 | 2ralbii 3163 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
16 | 9, 15 | bitr4i 279 | . . . . . 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 1429 | . . . 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 2818 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 10, 19, 11 | issubrg2 19484 | . . . 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 19308 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂) |
22 | 3, 19 | oppr1 19313 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑂) |
23 | 21, 22, 12 | issubrg2 19484 | . . . . 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 218 | . . . 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 312 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂))) |
26 | 1, 5, 25 | pm5.21nii 380 | . 2 ⊢ (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂)) |
27 | 26 | eqriv 2815 | 1 ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 SubGrpcsubg 18211 1rcur 19180 Ringcrg 19226 opprcoppr 19301 SubRingcsubrg 19460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-subg 18214 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-subrg 19462 |
This theorem is referenced by: (None) |
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