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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 18960 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| 8 | 1, 2, 3, 4, 7 | syl13anc 1374 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-sgrp 18732 df-mnd 18748 df-grp 18954 |
| This theorem is referenced by: grplmulf1o 19031 grpraddf1o 19032 eqger 19196 conjnmz 19270 psdmul 22170 rloccring 33274 qsdrngilem 33522 grpcominv1 42518 |
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