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| Mirrors > Home > MPE Home > Th. List > grpcld | Structured version Visualization version GIF version | ||
| Description: Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.) | 
| Ref | Expression | 
|---|---|
| grpcld.b | ⊢ 𝐵 = (Base‘𝐺) | 
| grpcld.p | ⊢ + = (+g‘𝐺) | 
| grpcld.r | ⊢ (𝜑 → 𝐺 ∈ Grp) | 
| grpcld.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| grpcld.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| grpcld | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grpcld.r | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
| 2 | grpcld.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | grpcld.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 4 | grpcld.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 5 | grpcld.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 6 | 4, 5 | grpcl 18959 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | 
| 7 | 1, 2, 3, 6 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 | 
| This theorem is referenced by: grpraddf1o 19032 dfgrp3 19057 xpsinv 19078 xpsgrpsub 19079 nmzsubg 19183 eqger 19196 conjnmz 19270 ghmqusnsg 19300 ghmquskerlem3 19304 lringuplu 20544 rnglidl1 21242 rngqiprngimfo 21311 rngqiprngfulem3 21323 mhpaddcl 22155 psdmul 22170 evls1addd 22375 evls1maprhm 22380 rhmmpl 22387 cphpyth 25250 ringdi22 33235 rlocaddval 33272 rloccring 33274 rlocf1 33277 evl1deg1 33601 evl1deg2 33602 evl1deg3 33603 ply1degltlss 33617 q1pdir 33623 r1pcyc 33627 r1padd1 33628 r1plmhm 33630 algextdeglem8 33765 rtelextdg2lem 33767 zrhcntr 33980 aks6d1c1p3 42111 aks5lem3a 42190 aks5lem5a 42192 grpcominv1 42518 rhmpsr 42562 mplmapghm 42566 evlsmaprhm 42580 evladdval 42585 selvadd 42598 | 
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