Theorem submcld 33174

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 18837	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
6	1, 2, 3, 5	syl3anc 1389	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6516  (class class class)co 7391  +_gcplusg 17277  SubMndcsubmnd 18807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-submnd 18809

This theorem is referenced by:  gsumwun  33217  rloccring  33413  rlocisunit  33418  ssdifidlprm  33606