Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscmgmALT | Structured version Visualization version GIF version |
Description: The predicate "is a commutative magma". (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 |
---|---|
iscmgmALT | ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6663 | . . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
2 | fveq2 6663 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
3 | 1, 2 | breq12d 5070 | . . 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 5066 | . . 3 ⊢ ( ⚬ comLaw 𝐵 ↔ (+g‘𝑀) comLaw (Base‘𝑀)) |
7 | 3, 6 | syl6bbr 290 | . 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ ⚬ comLaw 𝐵)) |
8 | df-cmgm2 44055 | . 2 ⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)} | |
9 | 7, 8 | elrab2 3680 | 1 ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 +gcplusg 16553 comLaw ccomlaw 44020 MgmALTcmgm2 44050 CMgmALTccmgm2 44051 |
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 |
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-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 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-iota 6307 df-fv 6356 df-cmgm2 44055 |
This theorem is referenced by: (None) |
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