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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 18906 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
8 | 1, 2, 3, 4, 7 | syl13anc 1369 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 Grpcgrp 18897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-sgrp 18686 df-mnd 18702 df-grp 18900 |
This theorem is referenced by: grplmulf1o 18976 grpraddf1o 18977 eqger 19140 conjnmz 19213 psdmul 22097 rloccring 33009 qsdrngilem 33230 grpcominv1 41779 |
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