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Mirrors > Home > MPE Home > Th. List > abl32 | Structured version Visualization version GIF version |
Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
ablcom.b | ⊢ 𝐵 = (Base‘𝐺) |
ablcom.p | ⊢ + = (+g‘𝐺) |
abl32.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
abl32.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
abl32.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
abl32.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
abl32 | ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abl32.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablcmn 18022 | . . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
4 | abl32.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | abl32.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | abl32.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
8 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | cmn32 18034 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
10 | 3, 4, 5, 6, 9 | syl13anc 1320 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 CMndccmn 18016 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-sgrp 17107 df-mnd 17118 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: matunitlindflem1 32575 baerlem5alem1 36015 baerlem5blem1 36016 |
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