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 6672 | . . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
2 | ismgmALT.o | . . . 4 ⊢ ⚬ = (+g‘𝑀) | |
3 | 1, 2 | syl6eqr 2876 | . . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ ) |
4 | fveq2 6672 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
5 | ismgmALT.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
6 | 4, 5 | syl6eqr 2876 | . . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
7 | 3, 6 | breq12d 5081 | . 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) clLaw (Base‘𝑚) ↔ ⚬ clLaw 𝐵)) |
8 | df-mgm2 44133 | . 2 ⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)} | |
9 | 7, 8 | elab2g 3670 | 1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 +gcplusg 16567 clLaw ccllaw 44097 MgmALTcmgm2 44129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-mgm2 44133 |
This theorem is referenced by: mgm2mgm 44141 |
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