Theorem submcld 18894

Description: Submonoids are closed under the monoid operation. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
submcld.1	⊢ + = (+g‘𝑀)
submcld.2	⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))
submcld.3	⊢ (𝜑 → 𝑋 ∈ 𝑆)
submcld.4	⊢ (𝜑 → 𝑌 ∈ 𝑆)

Assertion

Ref	Expression
submcld	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem submcld

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 18893	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
6	1, 2, 3, 5	syl3anc 1398	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7416  +_gcplusg 17327  SubMndcsubmnd 18863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-submnd 18865

This theorem is used by:  ssdifidlprm  21515  gsumwun  33419  rloccring  33614  rlocisunit  33619