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Theorem mnd32g 17226
Description: Commutative/associative law for 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)
mnd32g.5 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
Assertion
Ref Expression
mnd32g (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Proof of Theorem mnd32g
StepHypRef Expression
1 mnd32g.5 . . 3 (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
21oveq2d 6620 . 2 (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (𝑍 + 𝑌)))
3 mnd4g.1 . . 3 (𝜑𝐺 ∈ Mnd)
4 mnd4g.2 . . 3 (𝜑𝑋𝐵)
5 mnd4g.3 . . 3 (𝜑𝑌𝐵)
6 mnd4g.4 . . 3 (𝜑𝑍𝐵)
7 mndcl.b . . . 4 𝐵 = (Base‘𝐺)
8 mndcl.p . . . 4 + = (+g𝐺)
97, 8mndass 17223 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
103, 4, 5, 6, 9syl13anc 1325 . 2 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
117, 8mndass 17223 . . 3 ((𝐺 ∈ Mnd ∧ (𝑋𝐵𝑍𝐵𝑌𝐵)) → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌)))
123, 4, 6, 5, 11syl13anc 1325 . 2 (𝜑 → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌)))
132, 10, 123eqtr4d 2665 1 (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mndcmnd 17215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-sgrp 17205  df-mnd 17216
This theorem is referenced by:  cmn32  18132
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