Theorem submcld 33011

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 18717	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
6	1, 2, 3, 5	syl3anc 1373	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  +_gcplusg 17158  SubMndcsubmnd 18687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-submnd 18689

This theorem is referenced by:  gsumwun  33040  rloccring  33232  ssdifidlprm  33418