Theorem ofaddmndmap 45631

Description: The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)

Hypotheses

Ref	Expression
ofaddmndmap.r	⊢ 𝑅 = (Base‘𝑀)
ofaddmndmap.p	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
ofaddmndmap	⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉))

Proof of Theorem ofaddmndmap

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1189	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑀 ∈ Mnd)
2		simprl 767	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
3		simprr 769	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
4		ofaddmndmap.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑀)
5		ofaddmndmap.p	. . . . 5 ⊢ + = (+g‘𝑀)
6	4, 5	mndcl 18374	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 + 𝑦) ∈ 𝑅)
7	1, 2, 3, 6	syl3anc 1369	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 + 𝑦) ∈ 𝑅)
8		elmapi 8611	. . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
9	8	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐴:𝑉⟶𝑅)
10	9	3ad2ant3 1133	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴:𝑉⟶𝑅)
11		elmapi 8611	. . . . 5 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵:𝑉⟶𝑅)
12	11	adantl 481	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐵:𝑉⟶𝑅)
13	12	3ad2ant3 1133	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵:𝑉⟶𝑅)
14		simp2 1135	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑉 ∈ 𝑌)
15		inidm 4157	. . 3 ⊢ (𝑉 ∩ 𝑉) = 𝑉
16	7, 10, 13, 14, 14, 15	off 7542	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵):𝑉⟶𝑅)
17	4	fvexi 6782	. . 3 ⊢ 𝑅 ∈ V
18		elmapg 8602	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝑌) → ((𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉) ↔ (𝐴 ∘f + 𝐵):𝑉⟶𝑅))
19	17, 14, 18	sylancr 586	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉) ↔ (𝐴 ∘f + 𝐵):𝑉⟶𝑅))
20	16, 19	mpbird 256	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268   ∘_f cof 7522   ↑_m cmap 8589  Basecbs 16893  +_gcplusg 16943  Mndcmnd 18366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-1st 7817  df-2nd 7818  df-map 8591  df-mgm 18307  df-sgrp 18356  df-mnd 18367

This theorem is referenced by:  lincsumcl  45724