Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofaddmndmap Structured version   Visualization version   GIF version

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

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1057 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → 𝑀 ∈ Mnd)
2 simprl 790 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
3 simprr 792 . . . 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 17124 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑥𝑅𝑦𝑅) → (𝑥 + 𝑦) ∈ 𝑅)
71, 2, 3, 6syl3anc 1318 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥 + 𝑦) ∈ 𝑅)
8 elmapi 7765 . . . . 5 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
98adantr 480 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐴:𝑉𝑅)
1093ad2ant3 1077 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴:𝑉𝑅)
11 elmapi 7765 . . . . 5 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵:𝑉𝑅)
1211adantl 481 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐵:𝑉𝑅)
13123ad2ant3 1077 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵:𝑉𝑅)
14 simp2 1055 . . 3 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑉𝑌)
15 inidm 3784 . . 3 (𝑉𝑉) = 𝑉
167, 10, 13, 14, 14, 15off 6810 . 2 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 + 𝐵):𝑉𝑅)
17 fvex 6113 . . . 4 (Base‘𝑀) ∈ V
184, 17eqeltri 2684 . . 3 𝑅 ∈ V
19 elmapg 7757 . . 3 ((𝑅 ∈ V ∧ 𝑉𝑌) → ((𝐴𝑓 + 𝐵) ∈ (𝑅𝑚 𝑉) ↔ (𝐴𝑓 + 𝐵):𝑉𝑅))
2018, 14, 19sylancr 694 . 2 ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 + 𝐵) ∈ (𝑅𝑚 𝑉) ↔ (𝐴𝑓 + 𝐵):𝑉𝑅))
2116, 20mpbird 246 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ↑𝑚 cmap 7744  Basecbs 15695  +gcplusg 15768  Mndcmnd 17117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-1st 7059  df-2nd 7060  df-map 7746  df-mgm 17065  df-sgrp 17107  df-mnd 17118 This theorem is referenced by:  lincsumcl  42014
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