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Mirrors > Home > MPE Home > Th. List > oppgsubg | Structured version Visualization version GIF version |
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppggic.o | ⊢ 𝑂 = (oppg‘𝐺) |
Ref | Expression |
---|---|
oppgsubg | ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 17422 | . . 3 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | subgrcl 17422 | . . . 4 ⊢ (𝑥 ∈ (SubGrp‘𝑂) → 𝑂 ∈ Grp) | |
3 | oppggic.o | . . . . 5 ⊢ 𝑂 = (oppg‘𝐺) | |
4 | 3 | oppggrpb 17611 | . . . 4 ⊢ (𝐺 ∈ Grp ↔ 𝑂 ∈ Grp) |
5 | 2, 4 | sylibr 223 | . . 3 ⊢ (𝑥 ∈ (SubGrp‘𝑂) → 𝐺 ∈ Grp) |
6 | 3 | oppgsubm 17615 | . . . . . . 7 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂) |
7 | 6 | eleq2i 2680 | . . . . . 6 ⊢ (𝑥 ∈ (SubMnd‘𝐺) ↔ 𝑥 ∈ (SubMnd‘𝑂)) |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubMnd‘𝐺) ↔ 𝑥 ∈ (SubMnd‘𝑂))) |
9 | eqid 2610 | . . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | 3, 9 | oppginv 17612 | . . . . . . . 8 ⊢ (𝐺 ∈ Grp → (invg‘𝐺) = (invg‘𝑂)) |
11 | 10 | fveq1d 6105 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘𝑦) = ((invg‘𝑂)‘𝑦)) |
12 | 11 | eleq1d 2672 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (((invg‘𝐺)‘𝑦) ∈ 𝑥 ↔ ((invg‘𝑂)‘𝑦) ∈ 𝑥)) |
13 | 12 | ralbidv 2969 | . . . . 5 ⊢ (𝐺 ∈ Grp → (∀𝑦 ∈ 𝑥 ((invg‘𝐺)‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥)) |
14 | 8, 13 | anbi12d 743 | . . . 4 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ (SubMnd‘𝐺) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝐺)‘𝑦) ∈ 𝑥) ↔ (𝑥 ∈ (SubMnd‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥))) |
15 | 9 | issubg3 17435 | . . . 4 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubGrp‘𝐺) ↔ (𝑥 ∈ (SubMnd‘𝐺) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝐺)‘𝑦) ∈ 𝑥))) |
16 | eqid 2610 | . . . . . 6 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
17 | 16 | issubg3 17435 | . . . . 5 ⊢ (𝑂 ∈ Grp → (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑥 ∈ (SubMnd‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥))) |
18 | 4, 17 | sylbi 206 | . . . 4 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑥 ∈ (SubMnd‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ((invg‘𝑂)‘𝑦) ∈ 𝑥))) |
19 | 14, 15, 18 | 3bitr4d 299 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑥 ∈ (SubGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝑂))) |
20 | 1, 5, 19 | pm5.21nii 367 | . 2 ⊢ (𝑥 ∈ (SubGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝑂)) |
21 | 20 | eqriv 2607 | 1 ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 SubMndcsubmnd 17157 Grpcgrp 17245 invgcminusg 17246 SubGrpcsubg 17411 oppgcoppg 17598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-subg 17414 df-oppg 17599 |
This theorem is referenced by: lsmmod2 17912 lsmdisj2r 17921 |
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