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Mirrors > Home > MPE Home > Th. List > grpassd | Structured version Visualization version GIF version |
Description: A group operation is associative. (Contributed by SN, 29-Jan-2025.) |
Ref | Expression |
---|---|
grpassd.b | ⊢ 𝐵 = (Base‘𝐺) |
grpassd.p | ⊢ + = (+g‘𝐺) |
grpassd.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grpassd.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
grpassd.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
grpassd.3 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
grpassd | ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpassd.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | grpassd.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | grpassd.2 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | grpassd.3 | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | grpassd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
6 | grpassd.p | . . 3 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | grpass 18827 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
8 | 1, 2, 3, 4, 7 | syl13anc 1372 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +gcplusg 17196 Grpcgrp 18818 |
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-ext 2703 ax-nul 5306 |
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 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 df-sgrp 18609 df-mnd 18625 df-grp 18821 |
This theorem is referenced by: eqger 19057 qsdrngilem 32603 grpcominv1 41084 |
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