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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscsgrpALT | Structured version Visualization version GIF version |
Description: The predicate "is a commutative semigroup". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ismgmALT.b | ⊢ 𝐵 = (Base‘𝑀) |
ismgmALT.o | ⊢ ⚬ = (+g‘𝑀) |
Ref | Expression |
---|---|
iscsgrpALT | ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6843 | . . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
2 | fveq2 6843 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
3 | 1, 2 | breq12d 5119 | . . 3 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ (+g‘𝑀) comLaw (Base‘𝑀))) |
4 | ismgmALT.o | . . . 4 ⊢ ⚬ = (+g‘𝑀) | |
5 | ismgmALT.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
6 | 4, 5 | breq12i 5115 | . . 3 ⊢ ( ⚬ comLaw 𝐵 ↔ (+g‘𝑀) comLaw (Base‘𝑀)) |
7 | 3, 6 | bitr4di 289 | . 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ ⚬ comLaw 𝐵)) |
8 | df-csgrp2 46242 | . 2 ⊢ CSGrpALT = {𝑚 ∈ SGrpALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)} | |
9 | 7, 8 | elrab2 3649 | 1 ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 Basecbs 17088 +gcplusg 17138 comLaw ccomlaw 46205 SGrpALTcsgrp2 46237 CSGrpALTccsgrp2 46238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-csgrp2 46242 |
This theorem is referenced by: (None) |
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