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| Mirrors > Home > MPE Home > Th. List > grpinvcld | Structured version Visualization version GIF version | ||
| Description: A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.) |
| Ref | Expression |
|---|---|
| grpinvcld.b | ⊢ 𝐵 = (Base‘𝐺) |
| grpinvcld.n | ⊢ 𝑁 = (invg‘𝐺) |
| grpinvcld.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
| grpinvcld.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| grpinvcld | ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpinvcld.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
| 2 | grpinvcld.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | grpinvcld.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | grpinvcld.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
| 5 | 3, 4 | grpinvcl 18943 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
| 6 | 1, 2, 5 | syl2anc 583 | 1 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 Basecbs 17179 Grpcgrp 18889 invgcminusg 18890 |
| 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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-riota 7376 df-ov 7423 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 |
| This theorem is referenced by: grpraddf1o 18969 xpsinv 19015 eqger 19132 conjnmz 19205 ghmquskerlem1 19233 rngmneg1 20106 rngmneg2 20107 rngm2neg 20108 rngsubdi 20110 rngsubdir 20111 cntzsubrng 20503 lssvnegcl 20839 rloccring 32984 ghmqusnsglem1 33129 qsdrngilem 33205 r1padd1 33274 grpcominv1 41748 |
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