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Mirrors > Home > MPE Home > Th. List > submgmcl | Structured version Visualization version GIF version |
Description: Submagmas are closed under the magma operation. (Contributed by AV, 26-Feb-2020.) |
Ref | Expression |
---|---|
submgmcl.p | ⊢ + = (+g‘𝑀) |
Ref | Expression |
---|---|
submgmcl | ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submgmrcl 18655 | . . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) | |
2 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | submgmcl.p | . . . . . . . 8 ⊢ + = (+g‘𝑀) | |
4 | 2, 3 | issubmgm 18662 | . . . . . . 7 ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))) |
6 | 5 | ibi 267 | . . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)) |
7 | 6 | simprd 495 | . . . 4 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) |
8 | ovrspc2v 7446 | . . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | |
9 | 7, 8 | sylan2 592 | . . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ 𝑆 ∈ (SubMgm‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆) |
10 | 9 | ancoms 458 | . 2 ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 + 𝑌) ∈ 𝑆) |
11 | 10 | 3impb 1113 | 1 ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⊆ wss 3947 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 Mgmcmgm 18598 SubMgmcsubmgm 18651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-submgm 18653 |
This theorem is referenced by: resmgmhm 18671 mgmhmima 18675 |
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