Metamath Proof Explorer


Theorem cnfldadd

Description: The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 6-Oct-2015) (Revised by Thierry Arnoux, 17-Dec-2017)

Ref Expression
Assertion cnfldadd + = ( +g ‘ ℂfld )

Proof

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