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Metamath Proof Explorer

Theorem nmpropd

Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses nmpropd.1 ⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) )
nmpropd.2 ⊢ ( 𝜑 → ( +g ‘ 𝐾 ) = ( +g ‘ 𝐿 ) )
nmpropd.3 ⊢ ( 𝜑 → ( dist ‘ 𝐾 ) = ( dist ‘ 𝐿 ) )
Assertion nmpropd ( 𝜑 → ( norm ‘ 𝐾 ) = ( norm ‘ 𝐿 ) )

Proof

Step Hyp Ref Expression
1 nmpropd.1 ⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) )
2 nmpropd.2 ⊢ ( 𝜑 → ( +g ‘ 𝐾 ) = ( +g ‘ 𝐿 ) )
3 nmpropd.3 ⊢ ( 𝜑 → ( dist ‘ 𝐾 ) = ( dist ‘ 𝐿 ) )
4 eqidd ⊢ ( 𝜑 → 𝑥 = 𝑥 )
5 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) )
6 2 oveqdr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
7 5 1 6 grpidpropd ⊢ ( 𝜑 → ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐿 ) )
8 3 4 7 oveq123d ⊢ ( 𝜑 → ( 𝑥 ( dist ‘ 𝐾 ) ( 0g ‘ 𝐾 ) ) = ( 𝑥 ( dist ‘ 𝐿 ) ( 0g ‘ 𝐿 ) ) )
9 1 8 mpteq12dv ⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝐾 ) ↦ ( 𝑥 ( dist ‘ 𝐾 ) ( 0g ‘ 𝐾 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐿 ) ↦ ( 𝑥 ( dist ‘ 𝐿 ) ( 0g ‘ 𝐿 ) ) ) )
10 eqid ⊢ ( norm ‘ 𝐾 ) = ( norm ‘ 𝐾 )
11 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 eqid ⊢ ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐾 )
13 eqid ⊢ ( dist ‘ 𝐾 ) = ( dist ‘ 𝐾 )
14 10 11 12 13 nmfval ⊢ ( norm ‘ 𝐾 ) = ( 𝑥 ∈ ( Base ‘ 𝐾 ) ↦ ( 𝑥 ( dist ‘ 𝐾 ) ( 0g ‘ 𝐾 ) ) )
15 eqid ⊢ ( norm ‘ 𝐿 ) = ( norm ‘ 𝐿 )
16 eqid ⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
17 eqid ⊢ ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝐿 )
18 eqid ⊢ ( dist ‘ 𝐿 ) = ( dist ‘ 𝐿 )
19 15 16 17 18 nmfval ⊢ ( norm ‘ 𝐿 ) = ( 𝑥 ∈ ( Base ‘ 𝐿 ) ↦ ( 𝑥 ( dist ‘ 𝐿 ) ( 0g ‘ 𝐿 ) ) )
20 9 14 19 3eqtr4g ⊢ ( 𝜑 → ( norm ‘ 𝐾 ) = ( norm ‘ 𝐿 ) )