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Mirrors > Home > MPE Home > Th. List > mnd32g | 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) |
mnd32g.5 | ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) |
Ref | Expression |
---|---|
mnd32g | ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnd32g.5 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) | |
2 | 1 | oveq2d 7374 | . 2 ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (𝑍 + 𝑌))) |
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 18570 | . . 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 18570 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌))) |
12 | 3, 4, 6, 5, 11 | syl13anc 1373 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌))) |
13 | 2, 10, 12 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Mndcmnd 18561 |
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 5264 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-sgrp 18551 df-mnd 18562 |
This theorem is referenced by: cmn32 19587 |
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