Theorem submcld 33117

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 18737	. 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 2113  ‘cfv 6492  (class class class)co 7358  +_gcplusg 17177  SubMndcsubmnd 18707

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-submnd 18709

This theorem is referenced by:  gsumwun  33158  rloccring  33352  ssdifidlprm  33539