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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprnsg | Structured version Visualization version GIF version |
Description: Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.) |
Ref | Expression |
---|---|
oppreqg.o | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprnsg | ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppreqg.o | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
2 | 1 | opprsubg 20120 | . . . . 5 ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) |
3 | 2 | eleq2i 2825 | . . . 4 ⊢ (𝑔 ∈ (SubGrp‘𝑅) ↔ 𝑔 ∈ (SubGrp‘𝑂)) |
4 | 3 | anbi1i 624 | . . 3 ⊢ ((𝑔 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝑔 → (𝑦(+g‘𝑅)𝑥) ∈ 𝑔)) ↔ (𝑔 ∈ (SubGrp‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝑔 → (𝑦(+g‘𝑅)𝑥) ∈ 𝑔))) |
5 | eqid 2732 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | eqid 2732 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
7 | 5, 6 | isnsg2 19010 | . . 3 ⊢ (𝑔 ∈ (NrmSGrp‘𝑅) ↔ (𝑔 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝑔 → (𝑦(+g‘𝑅)𝑥) ∈ 𝑔))) |
8 | 1, 5 | opprbas 20111 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑂) |
9 | 1, 6 | oppradd 20113 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑂) |
10 | 8, 9 | isnsg2 19010 | . . 3 ⊢ (𝑔 ∈ (NrmSGrp‘𝑂) ↔ (𝑔 ∈ (SubGrp‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝑔 → (𝑦(+g‘𝑅)𝑥) ∈ 𝑔))) |
11 | 4, 7, 10 | 3bitr4i 302 | . 2 ⊢ (𝑔 ∈ (NrmSGrp‘𝑅) ↔ 𝑔 ∈ (NrmSGrp‘𝑂)) |
12 | 11 | eqriv 2729 | 1 ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 +gcplusg 17181 SubGrpcsubg 18974 NrmSGrpcnsg 18975 opprcoppr 20103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-tpos 8195 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-0g 17371 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-grp 18799 df-subg 18977 df-nsg 18978 df-oppr 20104 |
This theorem is referenced by: opprqusplusg 32513 |
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