Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18327 | . . . 4 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
4 | 3 | ibi 266 | . . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) |
5 | ovrspc2v 7301 | . . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | |
6 | 5 | expcom 414 | . . 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 1115 | 1 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 Mgmcmgm 18324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-mgm 18326 |
This theorem is referenced by: isnmgm 18330 mgmsscl 18331 mgmplusf 18336 issstrmgm 18337 gsummgmpropd 18365 mndcl 18393 gsumsgrpccat 18478 smndex1sgrp 18547 dfgrp2 18604 dfgrp3e 18675 mulgnncl 18719 mulgnndir 18732 mgmhmf1o 45341 idmgmhm 45342 issubmgm2 45344 rabsubmgmd 45345 mgmhmco 45355 mgmhmeql 45357 submgmacs 45358 mgmplusgiopALT 45388 rngcl 45441 c0mgm 45467 c0snmgmhm 45472 |
Copyright terms: Public domain | W3C validator |