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Theorem mnd4g 17288
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 17287 . . 3 (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊)))
98oveq2d 6651 . 2 (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
10 mnd4g.2 . . 3 (𝜑𝑋𝐵)
111, 2mndcl 17282 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑍𝐵𝑊𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
123, 5, 6, 11syl3anc 1324 . . 3 (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
131, 2mndass 17283 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
143, 10, 4, 12, 13syl13anc 1326 . 2 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
151, 2mndcl 17282 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑌𝐵𝑊𝐵) → (𝑌 + 𝑊) ∈ 𝐵)
163, 4, 6, 15syl3anc 1324 . . 3 (𝜑 → (𝑌 + 𝑊) ∈ 𝐵)
171, 2mndass 17283 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑍𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
183, 10, 5, 16, 17syl13anc 1326 . 2 (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
199, 14, 183eqtr4d 2664 1 (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Mndcmnd 17275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-mgm 17223  df-sgrp 17265  df-mnd 17276
This theorem is referenced by:  lsmsubm  18049  pj1ghm  18097  cmn4  18193  gsumzaddlem  18302
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