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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 19174 | . . 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 19186 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
10 | 3, 4, 5, 6, 9 | syl13anc 1374 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6377 (class class class)co 7210 Basecbs 16757 +gcplusg 16799 CMndccmn 19167 Abelcabl 19168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-nul 5196 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-sbc 3692 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-iota 6335 df-fv 6385 df-ov 7213 df-sgrp 18160 df-mnd 18171 df-cmn 19169 df-abl 19170 |
This theorem is referenced by: matunitlindflem1 35508 baerlem5alem1 39457 baerlem5blem1 39458 |
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