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| Mirrors > Home > MPE Home > Th. List > opprsubg | Structured version Visualization version GIF version | ||
| Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
| Ref | Expression |
|---|---|
| opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
| Ref | Expression |
|---|---|
| opprsubg | ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opprbas.1 | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
| 2 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 3 | 1, 2 | opprbas 20341 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
| 4 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 5 | 1, 4 | oppradd 20343 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑂) |
| 6 | 3, 5 | grpprop 18970 | . . . 4 ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp) |
| 7 | biid 261 | . . . 4 ⊢ (𝑥 ⊆ (Base‘𝑅) ↔ 𝑥 ⊆ (Base‘𝑅)) | |
| 8 | eqid 2737 | . . . . . . . 8 ⊢ (𝑅 ↾s 𝑥) = (𝑅 ↾s 𝑥) | |
| 9 | 8, 2 | ressbas 17280 | . . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝑥))) |
| 10 | 9 | elv 3485 | . . . . . 6 ⊢ (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝑥)) |
| 11 | eqid 2737 | . . . . . . . 8 ⊢ (𝑂 ↾s 𝑥) = (𝑂 ↾s 𝑥) | |
| 12 | 11, 3 | ressbas 17280 | . . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑂 ↾s 𝑥))) |
| 13 | 12 | elv 3485 | . . . . . 6 ⊢ (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑂 ↾s 𝑥)) |
| 14 | 10, 13 | eqtr3i 2767 | . . . . 5 ⊢ (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑂 ↾s 𝑥)) |
| 15 | 8, 4 | ressplusg 17334 | . . . . . . 7 ⊢ (𝑥 ∈ V → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝑥))) |
| 16 | 11, 5 | ressplusg 17334 | . . . . . . 7 ⊢ (𝑥 ∈ V → (+g‘𝑅) = (+g‘(𝑂 ↾s 𝑥))) |
| 17 | 15, 16 | eqtr3d 2779 | . . . . . 6 ⊢ (𝑥 ∈ V → (+g‘(𝑅 ↾s 𝑥)) = (+g‘(𝑂 ↾s 𝑥))) |
| 18 | 17 | elv 3485 | . . . . 5 ⊢ (+g‘(𝑅 ↾s 𝑥)) = (+g‘(𝑂 ↾s 𝑥)) |
| 19 | 14, 18 | grpprop 18970 | . . . 4 ⊢ ((𝑅 ↾s 𝑥) ∈ Grp ↔ (𝑂 ↾s 𝑥) ∈ Grp) |
| 20 | 6, 7, 19 | 3anbi123i 1156 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp)) |
| 21 | 2 | issubg 19144 | . . 3 ⊢ (𝑥 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp)) |
| 22 | 3 | issubg 19144 | . . 3 ⊢ (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp)) |
| 23 | 20, 21, 22 | 3bitr4i 303 | . 2 ⊢ (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂)) |
| 24 | 23 | eqriv 2734 | 1 ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 Grpcgrp 18951 SubGrpcsubg 19138 opprcoppr 20333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-subg 19141 df-oppr 20334 |
| This theorem is referenced by: opprsubrng 20559 opprsubrg 20593 isridlrng 21229 isridl 21262 opprnsg 33512 |
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