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Metamath Proof Explorer

Theorem cntzcmnf

Description: Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016)

Ref Expression
Hypotheses cntzcmnf.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
cntzcmnf.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
cntzcmnf.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
cntzcmnf.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
Assertion cntzcmnf ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )

Proof

Step Hyp Ref Expression
1 cntzcmnf.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 cntzcmnf.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
3 cntzcmnf.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4 cntzcmnf.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
5 4 frnd ⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐵 )
6 1 2 cntzcmn ⊢ ( ( 𝐺 ∈ CMnd ∧ ran 𝐹 ⊆ 𝐵 ) → ( 𝑍 ‘ ran 𝐹 ) = 𝐵 )
7 3 5 6 syl2anc ⊢ ( 𝜑 → ( 𝑍 ‘ ran 𝐹 ) = 𝐵 )
8 5 7 sseqtrrd ⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )