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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 17066 | . . . 4 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
4 | 3 | ibi 255 | . . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) |
5 | ovrspc2v 6571 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | |
6 | 5 | expcom 450 | . . 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 1254 | 1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Mgmcmgm 17063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-mgm 17065 |
This theorem is referenced by: isnmgm 17069 mgmplusf 17074 issstrmgm 17075 gsummgmpropd 17098 mndcl 17124 dfgrp2 17270 dfgrp3e 17338 mulgnncl 17379 mulgnndir 17392 mgmhmf1o 41577 idmgmhm 41578 issubmgm2 41580 rabsubmgmd 41581 mgmhmco 41591 mgmhmeql 41593 submgmacs 41594 mgmplusgiopALT 41620 rngcl 41673 c0mgm 41699 c0snmgmhm 41704 |
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