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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 17972 | . . . 4 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
4 | 3 | ibi 270 | . . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) |
5 | ovrspc2v 7199 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | |
6 | 5 | expcom 417 | . . 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 +gcplusg 16671 Mgmcmgm 17969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-iota 6298 df-fv 6348 df-ov 7176 df-mgm 17971 |
This theorem is referenced by: isnmgm 17975 mgmsscl 17976 mgmplusf 17981 issstrmgm 17982 gsummgmpropd 18010 mndcl 18038 gsumsgrpccat 18123 smndex1sgrp 18192 dfgrp2 18249 dfgrp3e 18320 mulgnncl 18364 mulgnndir 18377 mgmhmf1o 44905 idmgmhm 44906 issubmgm2 44908 rabsubmgmd 44909 mgmhmco 44919 mgmhmeql 44921 submgmacs 44922 mgmplusgiopALT 44952 rngcl 45005 c0mgm 45031 c0snmgmhm 45036 |
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