Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mnd4g Structured version   Visualization version   GIF version

Theorem mnd4g 17130
 Description: Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
mndcl.b 𝐵 = (Base‘𝐺)
mndcl.p + = (+g𝐺)
mnd4g.1 (𝜑𝐺 ∈ Mnd)
mnd4g.2 (𝜑𝑋𝐵)
mnd4g.3 (𝜑𝑌𝐵)
mnd4g.4 (𝜑𝑍𝐵)
mnd4g.5 (𝜑𝑊𝐵)
mnd4g.6 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
Assertion
Ref Expression
mnd4g (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Proof of Theorem mnd4g
StepHypRef Expression
1 mndcl.b . . . 4 𝐵 = (Base‘𝐺)
2 mndcl.p . . . 4 + = (+g𝐺)
3 mnd4g.1 . . . 4 (𝜑𝐺 ∈ Mnd)
4 mnd4g.3 . . . 4 (𝜑𝑌𝐵)
5 mnd4g.4 . . . 4 (𝜑𝑍𝐵)
6 mnd4g.5 . . . 4 (𝜑𝑊𝐵)
7 mnd4g.6 . . . 4 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
81, 2, 3, 4, 5, 6, 7mnd12g 17129 . . 3 (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊)))
98oveq2d 6565 . 2 (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
10 mnd4g.2 . . 3 (𝜑𝑋𝐵)
111, 2mndcl 17124 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
123, 5, 6, 11syl3anc 1318 . . 3 (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
131, 2mndass 17125 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
143, 10, 4, 12, 13syl13anc 1320 . 2 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
151, 2mndcl 17124 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑌𝐵𝑊𝐵) → (𝑌 + 𝑊) ∈ 𝐵)
163, 4, 6, 15syl3anc 1318 . . 3 (𝜑 → (𝑌 + 𝑊) ∈ 𝐵)
171, 2mndass 17125 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
183, 10, 5, 16, 17syl13anc 1320 . 2 (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
199, 14, 183eqtr4d 2654 1 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-mgm 17065  df-sgrp 17107  df-mnd 17118 This theorem is referenced by:  lsmsubm  17891  pj1ghm  17939  cmn4  18035  gsumzaddlem  18144
 Copyright terms: Public domain W3C validator