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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > submcld | Structured version Visualization version GIF version |
Description: Submonoids are closed under the monoid operation. (Contributed by Thierry Arnoux, 4-May-2025.) |
Ref | Expression |
---|---|
submcld.1 | ⊢ + = (+g‘𝑀) |
submcld.2 | ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀)) |
submcld.3 | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
submcld.4 | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
submcld | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submcld.2 | . 2 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀)) | |
2 | submcld.3 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
3 | submcld.4 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
4 | submcld.1 | . . 3 ⊢ + = (+g‘𝑀) | |
5 | 4 | submcl 18838 | . 2 ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |
6 | 1, 2, 3, 5 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 +gcplusg 17298 SubMndcsubmnd 18808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-submnd 18810 |
This theorem is referenced by: gsumwun 33051 rloccring 33257 ssdifidlprm 33466 |
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