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Theorem ofaddmndmap 44672
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 + = (+g𝑀)
Assertion
Ref Expression
ofaddmndmap ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f + 𝐵) ∈ (𝑅m 𝑉))

Proof of Theorem ofaddmndmap
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1188 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → 𝑀 ∈ Mnd)
2 simprl 770 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
3 simprr 772 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
4 ofaddmndmap.r . . . . 5 𝑅 = (Base‘𝑀)
5 ofaddmndmap.p . . . . 5 + = (+g𝑀)
64, 5mndcl 17922 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑥𝑅𝑦𝑅) → (𝑥 + 𝑦) ∈ 𝑅)
71, 2, 3, 6syl3anc 1368 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥 + 𝑦) ∈ 𝑅)
8 elmapi 8425 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
98adantr 484 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴:𝑉𝑅)
1093ad2ant3 1132 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴:𝑉𝑅)
11 elmapi 8425 . . . . 5 (𝐵 ∈ (𝑅m 𝑉) → 𝐵:𝑉𝑅)
1211adantl 485 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵:𝑉𝑅)
13123ad2ant3 1132 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵:𝑉𝑅)
14 simp2 1134 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑉𝑌)
15 inidm 4181 . . 3 (𝑉𝑉) = 𝑉
167, 10, 13, 14, 14, 15off 7419 . 2 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f + 𝐵):𝑉𝑅)
174fvexi 6676 . . 3 𝑅 ∈ V
18 elmapg 8416 . . 3 ((𝑅 ∈ V ∧ 𝑉𝑌) → ((𝐴f + 𝐵) ∈ (𝑅m 𝑉) ↔ (𝐴f + 𝐵):𝑉𝑅))
1917, 14, 18sylancr 590 . 2 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f + 𝐵) ∈ (𝑅m 𝑉) ↔ (𝐴f + 𝐵):𝑉𝑅))
2016, 19mpbird 260 1 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f + 𝐵) ∈ (𝑅m 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3481  wf 6340  cfv 6344  (class class class)co 7150  f cof 7402  m cmap 8403  Basecbs 16486  +gcplusg 16568  Mndcmnd 17914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7404  df-1st 7685  df-2nd 7686  df-map 8405  df-mgm 17855  df-sgrp 17904  df-mnd 17915
This theorem is referenced by:  lincsumcl  44767
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