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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 19522 | . . 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 19535 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
10 | 3, 4, 5, 6, 9 | syl13anc 1372 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 Basecbs 17037 +gcplusg 17087 CMndccmn 19515 Abelcabl 19516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2707 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7354 df-sgrp 18500 df-mnd 18511 df-cmn 19517 df-abl 19518 |
This theorem is referenced by: matunitlindflem1 36006 baerlem5alem1 40103 baerlem5blem1 40104 |
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