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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 18869 | . 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 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Grpcgrp 18861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 |
This theorem is referenced by: dfgrp3 18965 xpsinv 18986 xpsgrpsub 18987 nmzsubg 19088 eqger 19101 lringuplu 20440 rnglidl1 21122 rngqiprngimfo 21149 rngqiprngfulem3 21161 cphpyth 25064 ghmquskerlem3 32971 evls1addd 33088 ply1degltlss 33108 q1pdir 33114 r1pcyc 33118 r1padd1 33119 r1plmhm 33121 evls1maprhm 33214 algextdeglem8 33235 grpcominv1 41549 rhmmpl 41588 mplmapghm 41591 evlsmaprhm 41605 evladdval 41610 selvadd 41623 |
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