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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 6665 | . 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 17302 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
10 | 3, 4, 5, 6, 9 | syl13anc 1328 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
11 | 7, 8 | mndass 17302 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) |
12 | 3, 5, 4, 6, 11 | syl13anc 1328 | . 2 ⊢ (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍))) |
13 | 2, 10, 12 | 3eqtr3d 2664 | 1 ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Mndcmnd 17294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: mnd4g 17307 cmn12 18213 |
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