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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 6645 | . . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
2 | fveq2 6645 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
3 | 1, 2 | breq12d 5043 | . . 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 5039 | . . 3 ⊢ ( ⚬ comLaw 𝐵 ↔ (+g‘𝑀) comLaw (Base‘𝑀)) |
7 | 3, 6 | syl6bbr 292 | . 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ ⚬ comLaw 𝐵)) |
8 | df-csgrp2 44482 | . 2 ⊢ CSGrpALT = {𝑚 ∈ SGrpALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)} | |
9 | 7, 8 | elrab2 3631 | 1 ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 +gcplusg 16557 comLaw ccomlaw 44445 SGrpALTcsgrp2 44477 CSGrpALTccsgrp2 44478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-csgrp2 44482 |
This theorem is referenced by: (None) |
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