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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 7447 | . 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 18756 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| 10 | 3, 4, 5, 6, 9 | syl13anc 1374 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| 11 | 7, 8 | mndass 18756 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌))) |
| 12 | 3, 4, 6, 5, 11 | syl13anc 1374 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌))) |
| 13 | 2, 10, 12 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mndcmnd 18747 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-sgrp 18732 df-mnd 18748 |
| This theorem is referenced by: cmn32 19818 gsumwun 33068 |
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