Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 485 | . 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 2818 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | eqid 2818 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 2, 3 | issgrp 17890 | . 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 6676 | . . 3 ⊢ (+g‘𝐺) ∈ V | |
6 | fvex 6676 | . . 3 ⊢ (Base‘𝐺) ∈ V | |
7 | isasslaw 44027 | . . 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 688 | . 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 293 | 1 ⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mgmcmgm 17838 Smgrpcsgrp 17888 assLaw casslaw 44019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-iota 6307 df-fv 6356 df-ov 7148 df-sgrp 17889 df-asslaw 44023 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |