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Mirrors > Home > MPE Home > Th. List > oppginv | Structured version Visualization version GIF version |
Description: Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
oppgbas.1 | ⊢ 𝑂 = (oppg‘𝑅) |
oppginv.2 | ⊢ 𝐼 = (invg‘𝑅) |
Ref | Expression |
---|---|
oppginv | ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | oppginv.2 | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
3 | 1, 2 | grpinvf 18150 | . . 3 ⊢ (𝑅 ∈ Grp → 𝐼:(Base‘𝑅)⟶(Base‘𝑅)) |
4 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | oppgbas.1 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
6 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
7 | 4, 5, 6 | oppgplus 18477 | . . . . 5 ⊢ ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(𝐼‘𝑥)) |
8 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 1, 4, 8, 2 | grprinv 18153 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(𝐼‘𝑥)) = (0g‘𝑅)) |
10 | 7, 9 | syl5eq 2868 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
11 | 10 | ralrimiva 3182 | . . 3 ⊢ (𝑅 ∈ Grp → ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
12 | 5 | oppggrp 18485 | . . . 4 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
13 | 5, 1 | oppgbas 18479 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
14 | 5, 8 | oppgid 18484 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑂) |
15 | eqid 2821 | . . . . 5 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
16 | 13, 6, 14, 15 | isgrpinv 18156 | . . . 4 ⊢ (𝑂 ∈ Grp → ((𝐼:(Base‘𝑅)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) ↔ (invg‘𝑂) = 𝐼)) |
17 | 12, 16 | syl 17 | . . 3 ⊢ (𝑅 ∈ Grp → ((𝐼:(Base‘𝑅)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) ↔ (invg‘𝑂) = 𝐼)) |
18 | 3, 11, 17 | mpbi2and 710 | . 2 ⊢ (𝑅 ∈ Grp → (invg‘𝑂) = 𝐼) |
19 | 18 | eqcomd 2827 | 1 ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 invgcminusg 18104 oppgcoppg 18473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-oppg 18474 |
This theorem is referenced by: oppgsubg 18491 oppgtgp 22706 tgpconncomp 22721 |
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