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Metamath Proof Explorer

Theorem mpocnfldadd

Description: The addition operation of the field of complex numbers. Version of cnfldadd using maps-to notation, which does not require ax-addf . (Contributed by GG, 31-Mar-2025)

Ref Expression
Assertion mpocnfldadd ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) = ( +g ‘ ℂfld )

Proof

Step Hyp Ref Expression
1 mpoaddex ⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ∈ V
2 cnfldstr ⊢ ℂfld Struct ⟨ 1 , 1 3 ⟩
3 plusgid ⊢ +g = Slot ( +g ‘ ndx )
4 snsstp2 ⊢ { ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ }
5 ssun1 ⊢ { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
6 ssun1 ⊢ ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ) ⊆ ( ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ } ) )
7 df-cnfld ⊢ ℂfld = ( ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ ( abs ∘ − ) ) ⟩ , ⟨ ( le ‘ ndx ) , ≤ ⟩ , ⟨ ( dist ‘ ndx ) , ( abs ∘ − ) ⟩ } ∪ { ⟨ ( UnifSet ‘ ndx ) , ( metUnif ‘ ( abs ∘ − ) ) ⟩ } ) )
8 6 7 sseqtrri ⊢ ( { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ) ⊆ ℂfld
9 5 8 sstri ⊢ { ⟨ ( Base ‘ ndx ) , ℂ ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ⟩ } ⊆ ℂfld
10 4 9 sstri ⊢ { ⟨ ( +g ‘ ndx ) , ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ⟩ } ⊆ ℂfld
11 2 3 10 strfv ⊢ ( ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) ∈ V → ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) = ( +g ‘ ℂfld ) )
12 1 11 ax-mp ⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + 𝑦 ) ) = ( +g ‘ ℂfld )