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Mirrors > Home > MPE Home > Th. List > mnd4g | Structured version Visualization version GIF version |
Description: Commutative/associative law for commutative 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) |
mnd4g.5 | ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
mnd4g.6 | ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) |
Ref | Expression |
---|---|
mnd4g | ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndcl.p | . . . 4 ⊢ + = (+g‘𝐺) | |
3 | mnd4g.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
4 | mnd4g.3 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | mnd4g.4 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
6 | mnd4g.5 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
7 | mnd4g.6 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mnd12g 18773 | . . 3 ⊢ (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊))) |
9 | 8 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
10 | mnd4g.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2 | mndcl 18768 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵) |
12 | 3, 5, 6, 11 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵) |
13 | 1, 2 | mndass 18769 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
14 | 3, 10, 4, 12, 13 | syl13anc 1371 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
15 | 1, 2 | mndcl 18768 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 + 𝑊) ∈ 𝐵) |
16 | 3, 4, 6, 15 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑊) ∈ 𝐵) |
17 | 1, 2 | mndass 18769 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
18 | 3, 10, 5, 16, 17 | syl13anc 1371 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
19 | 9, 14, 18 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Mndcmnd 18760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-mgm 18666 df-sgrp 18745 df-mnd 18761 |
This theorem is referenced by: lsmsubm 19686 pj1ghm 19736 cmn4 19834 gsumzaddlem 19954 |
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