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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 2727 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | oppginv.2 | . . . 4 ⊢ 𝐼 = (invg‘𝑅) | |
3 | 1, 2 | grpinvf 18928 | . . 3 ⊢ (𝑅 ∈ Grp → 𝐼:(Base‘𝑅)⟶(Base‘𝑅)) |
4 | eqid 2727 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | oppgbas.1 | . . . . . 6 ⊢ 𝑂 = (oppg‘𝑅) | |
6 | eqid 2727 | . . . . . 6 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
7 | 4, 5, 6 | oppgplus 19284 | . . . . 5 ⊢ ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (𝑥(+g‘𝑅)(𝐼‘𝑥)) |
8 | eqid 2727 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 1, 4, 8, 2 | grprinv 18932 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(𝐼‘𝑥)) = (0g‘𝑅)) |
10 | 7, 9 | eqtrid 2779 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
11 | 10 | ralrimiva 3141 | . . 3 ⊢ (𝑅 ∈ Grp → ∀𝑥 ∈ (Base‘𝑅)((𝐼‘𝑥)(+g‘𝑂)𝑥) = (0g‘𝑅)) |
12 | 5 | oppggrp 19295 | . . . 4 ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp) |
13 | 5, 1 | oppgbas 19287 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
14 | 5, 8 | oppgid 19294 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑂) |
15 | eqid 2727 | . . . . 5 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
16 | 13, 6, 14, 15 | isgrpinv 18935 | . . . 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 711 | . 2 ⊢ (𝑅 ∈ Grp → (invg‘𝑂) = 𝐼) |
19 | 18 | eqcomd 2733 | 1 ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 +gcplusg 17218 0gc0g 17406 Grpcgrp 18875 invgcminusg 18876 oppgcoppg 19280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-minusg 18879 df-oppg 19281 |
This theorem is referenced by: oppgsubg 19301 oppgtgp 23976 tgpconncomp 23991 |
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