Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20057 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | subrgrcl 20057 | . . . 4 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑂 ∈ Ring) | |
3 | opprsubrg.o | . . . . 5 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | 3 | opprringb 19902 | . . . 4 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring) |
5 | 2, 4 | sylibr 233 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑅 ∈ Ring) |
6 | 3 | opprsubg 19906 | . . . . . . 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 3260 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) | |
10 | eqid 2733 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2733 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | eqid 2733 | . . . . . . . . . 10 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
13 | 10, 11, 3, 12 | opprmul 19893 | . . . . . . . . 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 3121 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
16 | 9, 15 | bitr4i 277 | . . . . . 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 1436 | . . . 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 2733 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 10, 19, 11 | issubrg2 20072 | . . . 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 19897 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂) |
22 | 3, 19 | oppr1 19904 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑂) |
23 | 21, 22, 12 | issubrg2 20072 | . . . . 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 216 | . . . 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 310 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂))) |
26 | 1, 5, 25 | pm5.21nii 379 | . 2 ⊢ (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂)) |
27 | 26 | eqriv 2730 | 1 ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1085 = wceq 1537 ∈ wcel 2101 ∀wral 3059 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 .rcmulr 16991 SubGrpcsubg 18777 1rcur 19765 Ringcrg 19811 opprcoppr 19889 SubRingcsubrg 20048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-2nd 7852 df-tpos 8062 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-0g 17180 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-grp 18608 df-subg 18780 df-mgp 19749 df-ur 19766 df-ring 19813 df-oppr 19890 df-subrg 20050 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |