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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 18191 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
7 | 1, 2, 3, 6 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6340 (class class class)co 7156 Basecbs 16555 +gcplusg 16637 Grpcgrp 18183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-nul 5180 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-iota 6299 df-fv 6348 df-ov 7159 df-mgm 17932 df-sgrp 17981 df-mnd 17992 df-grp 18186 |
This theorem is referenced by: (None) |
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