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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 7160 | . 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 17908 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
10 | 3, 4, 5, 6, 9 | syl13anc 1364 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
11 | 7, 8 | mndass 17908 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) |
12 | 3, 5, 4, 6, 11 | syl13anc 1364 | . 2 ⊢ (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) |
13 | 2, 10, 12 | 3eqtr3d 2861 | 1 ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mndcmnd 17899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-sgrp 17889 df-mnd 17900 |
This theorem is referenced by: mnd4g 17913 cmn12 18856 |
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