![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mnd12g | Structured version Visualization version GIF version |
Description: Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
mndcl.b | ⊢ 𝐵 = (Base‘𝐺) |
mndcl.p | ⊢ + = (+g‘𝐺) |
mnd4g.1 | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
mnd4g.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
mnd4g.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
mnd4g.4 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
mnd12g.5 | ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Ref | Expression |
---|---|
mnd12g | ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnd12g.5 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | |
2 | 1 | oveq1d 7424 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑌 + 𝑋) + 𝑍)) |
3 | mnd4g.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
4 | mnd4g.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | mnd4g.3 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | mnd4g.4 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | mndcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
8 | mndcl.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | mndass 18634 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
10 | 3, 4, 5, 6, 9 | syl13anc 1373 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
11 | 7, 8 | mndass 18634 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) |
12 | 3, 5, 4, 6, 11 | syl13anc 1373 | . 2 ⊢ (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) |
13 | 2, 10, 12 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 Mndcmnd 18625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-sgrp 18610 df-mnd 18626 |
This theorem is referenced by: mnd4g 18639 cmn12 19670 |
Copyright terms: Public domain | W3C validator |