Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ismgmALT | Structured version Visualization version GIF version |
Description: The predicate "is a magma". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ismgmALT.b | ⊢ 𝐵 = (Base‘𝑀) |
ismgmALT.o | ⊢ ⚬ = (+g‘𝑀) |
Ref | Expression |
---|---|
ismgmALT | ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6839 | . . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
2 | ismgmALT.o | . . . 4 ⊢ ⚬ = (+g‘𝑀) | |
3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ ) |
4 | fveq2 6839 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
5 | ismgmALT.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
6 | 4, 5 | eqtr4di 2795 | . . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
7 | 3, 6 | breq12d 5116 | . 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) clLaw (Base‘𝑚) ↔ ⚬ clLaw 𝐵)) |
8 | df-mgm2 46054 | . 2 ⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)} | |
9 | 7, 8 | elab2g 3630 | 1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 Basecbs 17043 +gcplusg 17093 clLaw ccllaw 46018 MgmALTcmgm2 46050 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-mgm2 46054 |
This theorem is referenced by: mgm2mgm 46062 |
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