Theorem ofaddmndmap 48068

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 1191	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑀 ∈ Mnd)
2		simprl 770	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
3		simprr 772	. . . 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 18780	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 + 𝑦) ∈ 𝑅)
7	1, 2, 3, 6	syl3anc 1371	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 + 𝑦) ∈ 𝑅)
8		elmapi 8907	. . . . 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 1135	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴:𝑉⟶𝑅)
11		elmapi 8907	. . . . 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 1135	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵:𝑉⟶𝑅)
14		simp2 1137	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑉 ∈ 𝑌)
15		inidm 4248	. . 3 ⊢ (𝑉 ∩ 𝑉) = 𝑉
16	7, 10, 13, 14, 14, 15	off 7732	. 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵):𝑉⟶𝑅)
17	4	fvexi 6934	. . 3 ⊢ 𝑅 ∈ V
18		elmapg 8897	. . 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 257	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∘_f cof 7712   ↑_m cmap 8884  Basecbs 17258  +_gcplusg 17311  Mndcmnd 18772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-1st 8030  df-2nd 8031  df-map 8886  df-mgm 18678  df-sgrp 18757  df-mnd 18773

This theorem is referenced by:  lincsumcl  48160