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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 2726 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | oppginv.2 | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
3 | 1, 2 | grpinvf 18916 | . . 3 ⊢ (𝑅 ∈ Grp → 𝐼:(Base‘𝑅)⟶(Base‘𝑅)) |
4 | eqid 2726 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | oppgbas.1 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
6 | eqid 2726 | . . . . . 6 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
7 | 4, 5, 6 | oppgplus 19265 | . . . . 5 ⊢ ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(𝐼‘𝑥)) |
8 | eqid 2726 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 1, 4, 8, 2 | grprinv 18920 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(𝐼‘𝑥)) = (0g‘𝑅)) |
10 | 7, 9 | eqtrid 2778 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
11 | 10 | ralrimiva 3140 | . . 3 ⊢ (𝑅 ∈ Grp → ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
12 | 5 | oppggrp 19276 | . . . 4 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
13 | 5, 1 | oppgbas 19268 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
14 | 5, 8 | oppgid 19275 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑂) |
15 | eqid 2726 | . . . . 5 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
16 | 13, 6, 14, 15 | isgrpinv 18923 | . . . 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 709 | . 2 ⊢ (𝑅 ∈ Grp → (invg‘𝑂) = 𝐼) |
19 | 18 | eqcomd 2732 | 1 ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 0gc0g 17394 Grpcgrp 18863 invgcminusg 18864 oppgcoppg 19261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-oppg 19262 |
This theorem is referenced by: oppgsubg 19282 oppgtgp 23957 tgpconncomp 23972 |
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