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Metamath Proof Explorer

Theorem ismgm

Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009) (Revised by AV, 6-Jan-2020)

Ref Expression
Hypotheses ismgm.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
ismgm.o ⊢ ⚬ = ( +g ‘ 𝑀 )
Assertion ismgm ( 𝑀 ∈ 𝑉 → ( 𝑀 ∈ Mgm ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ismgm.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
2 ismgm.o ⊢ ⚬ = ( +g ‘ 𝑀 )
3 fvexd ⊢ ( 𝑚 = 𝑀 → ( Base ‘ 𝑚 ) ∈ V )
4 fveq2 ⊢ ( 𝑚 = 𝑀 → ( Base ‘ 𝑚 ) = ( Base ‘ 𝑀 ) )
5 4 1 eqtr4di ⊢ ( 𝑚 = 𝑀 → ( Base ‘ 𝑚 ) = 𝐵 )
6 fvexd ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) → ( +g ‘ 𝑚 ) ∈ V )
7 fveq2 ⊢ ( 𝑚 = 𝑀 → ( +g ‘ 𝑚 ) = ( +g ‘ 𝑀 ) )
8 7 adantr ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) → ( +g ‘ 𝑚 ) = ( +g ‘ 𝑀 ) )
9 8 2 eqtr4di ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) → ( +g ‘ 𝑚 ) = ⚬ )
10 simplr ⊢ ( ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵 )
11 oveq ⊢ ( 𝑜 = ⚬ → ( 𝑥 𝑜 𝑦 ) = ( 𝑥 ⚬ 𝑦 ) )
12 11 adantl ⊢ ( ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) ∧ 𝑜 = ⚬ ) → ( 𝑥 𝑜 𝑦 ) = ( 𝑥 ⚬ 𝑦 ) )
13 12 10 eleq12d ⊢ ( ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) ∧ 𝑜 = ⚬ ) → ( ( 𝑥 𝑜 𝑦 ) ∈ 𝑏 ↔ ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )
14 10 13 raleqbidv ⊢ ( ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) ∧ 𝑜 = ⚬ ) → ( ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑜 𝑦 ) ∈ 𝑏 ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )
15 10 14 raleqbidv ⊢ ( ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) ∧ 𝑜 = ⚬ ) → ( ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑜 𝑦 ) ∈ 𝑏 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )
16 6 9 15 sbcied2 ⊢ ( ( 𝑚 = 𝑀 ∧ 𝑏 = 𝐵 ) → ( [ ( +g ‘ 𝑚 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑜 𝑦 ) ∈ 𝑏 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )
17 3 5 16 sbcied2 ⊢ ( 𝑚 = 𝑀 → ( [ ( Base ‘ 𝑚 ) / 𝑏 ] [ ( +g ‘ 𝑚 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑜 𝑦 ) ∈ 𝑏 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )
18 df-mgm ⊢ Mgm = { 𝑚 ∣ [ ( Base ‘ 𝑚 ) / 𝑏 ] [ ( +g ‘ 𝑚 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑜 𝑦 ) ∈ 𝑏 }
19 17 18 elab2g ⊢ ( 𝑀 ∈ 𝑉 → ( 𝑀 ∈ Mgm ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ⚬ 𝑦 ) ∈ 𝐵 ) )