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| Mirrors > Home > MPE Home > Th. List > mgmcl | Structured version Visualization version GIF version | ||
| Description: Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.) |
| Ref | Expression |
|---|---|
| mgmcl.b | ⊢ 𝐵 = (Base‘𝑀) |
| mgmcl.o | ⊢ ⚬ = (+g‘𝑀) |
| Ref | Expression |
|---|---|
| mgmcl | ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mgmcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
| 2 | mgmcl.o | . . . . 5 ⊢ ⚬ = (+g‘𝑀) | |
| 3 | 1, 2 | ismgm 18654 | . . . 4 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
| 4 | 3 | ibi 267 | . . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) |
| 5 | ovrspc2v 7457 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | |
| 6 | 5 | expcom 413 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)) |
| 7 | 4, 6 | syl 17 | . 2 ⊢ (𝑀 ∈ Mgm → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)) |
| 8 | 7 | 3impib 1117 | 1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 |
| 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-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 |
| This theorem is referenced by: isnmgm 18657 mgmsscl 18658 mgmplusf 18663 issstrmgm 18666 gsummgmpropd 18694 mgmhmf1o 18713 idmgmhm 18714 issubmgm2 18716 rabsubmgmd 18717 mgmhmco 18727 mgmhmeql 18729 submgmacs 18730 sgrpcl 18739 mndcl 18755 gsumsgrpccat 18853 smndex1sgrp 18921 dfgrp2 18980 dfgrp3e 19058 mulgnncl 19107 mulgnndir 19121 rngcl 20161 c0mgm 20459 c0snmgmhm 20462 psraddcl 21958 mgmplusgiopALT 48110 |
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