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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 20593 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | subrgrcl 20593 | . . . 4 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑂 ∈ Ring) | |
3 | opprsubrg.o | . . . . 5 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | 3 | opprringb 20365 | . . . 4 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring) |
5 | 2, 4 | sylibr 234 | . . 3 ⊢ (𝑥 ∈ (SubRing‘𝑂) → 𝑅 ∈ Ring) |
6 | 3 | opprsubg 20369 | . . . . . . 7 ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) = (SubGrp‘𝑂)) |
8 | 7 | eleq2d 2825 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂))) |
9 | ralcom 3287 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) | |
10 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2735 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | eqid 2735 | . . . . . . . . . 10 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
13 | 10, 11, 3, 12 | opprmul 20354 | . . . . . . . . 9 ⊢ (𝑧(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑧) |
14 | 13 | eleq1i 2830 | . . . . . . . 8 ⊢ ((𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
15 | 14 | 2ralbii 3126 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑂)𝑦) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑦(.r‘𝑅)𝑧) ∈ 𝑥) |
16 | 9, 15 | bitr4i 278 | . . . . . 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 1437 | . . . 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 2735 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
20 | 10, 19, 11 | issubrg2 20609 | . . . 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 20358 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂) |
22 | 3, 19 | oppr1 20367 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑂) |
23 | 21, 22, 12 | issubrg2 20609 | . . . . 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 217 | . . . 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 311 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂))) |
26 | 1, 5, 25 | pm5.21nii 378 | . 2 ⊢ (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ (SubRing‘𝑂)) |
27 | 26 | eqriv 2732 | 1 ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 SubGrpcsubg 19151 1rcur 20199 Ringcrg 20251 opprcoppr 20350 SubRingcsubrg 20586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-subrng 20563 df-subrg 20587 |
This theorem is referenced by: (None) |
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