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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 18972 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
7 | 1, 2, 3, 6 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 |
This theorem is referenced by: grpraddf1o 19045 dfgrp3 19070 xpsinv 19091 xpsgrpsub 19092 nmzsubg 19196 eqger 19209 conjnmz 19283 ghmqusnsg 19313 ghmquskerlem3 19317 lringuplu 20561 rnglidl1 21260 rngqiprngimfo 21329 rngqiprngfulem3 21341 mhpaddcl 22173 psdmul 22188 evls1addd 22391 evls1maprhm 22396 rhmmpl 22403 cphpyth 25264 ringdi22 33221 rlocaddval 33255 rloccring 33257 rlocf1 33260 evl1deg1 33581 evl1deg2 33582 evl1deg3 33583 ply1degltlss 33597 q1pdir 33603 r1pcyc 33607 r1padd1 33608 r1plmhm 33610 algextdeglem8 33730 rtelextdg2lem 33732 zrhcntr 33942 aks6d1c1p3 42092 aks5lem3a 42171 aks5lem5a 42173 grpcominv1 42495 rhmpsr 42539 mplmapghm 42543 evlsmaprhm 42557 evladdval 42562 selvadd 42575 |
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