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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgrpplusgaopALT | Structured version Visualization version GIF version |
Description: Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sgrpplusgaopALT | ⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . 2 ⊢ ((𝐺 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) | |
2 | eqid 2734 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | eqid 2734 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 2, 3 | issgrp 18745 | . 2 ⊢ (𝐺 ∈ Smgrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))) |
5 | fvex 6919 | . . 3 ⊢ (+g‘𝐺) ∈ V | |
6 | fvex 6919 | . . 3 ⊢ (Base‘𝐺) ∈ V | |
7 | isasslaw 48035 | . . 3 ⊢ (((+g‘𝐺) ∈ V ∧ (Base‘𝐺) ∈ V) → ((+g‘𝐺) assLaw (Base‘𝐺) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))) | |
8 | 5, 6, 7 | mp2an 692 | . 2 ⊢ ((+g‘𝐺) assLaw (Base‘𝐺) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) |
9 | 1, 4, 8 | 3imtr4i 292 | 1 ⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 Mgmcmgm 18663 Smgrpcsgrp 18743 assLaw casslaw 48027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-iota 6515 df-fv 6570 df-ov 7433 df-sgrp 18744 df-asslaw 48031 |
This theorem is referenced by: (None) |
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