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Metamath Proof Explorer

Theorem sn-mul02

Description: mul02 without ax-mulcom . See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md for an outline. (Contributed by SN, 30-Jun-2024)

Ref Expression
Assertion sn-mul02 ( 𝐴 ∈ ℂ → ( 0 · 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 cnre ⊢ ( 𝐴 ∈ ℂ → ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝐴 = ( 𝑥 + ( i · 𝑦 ) ) )
2 0cnd ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 0 ∈ ℂ )
3 recn ⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℂ )
4 3 adantr ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 𝑥 ∈ ℂ )
5 ax-icn ⊢ i ∈ ℂ
6 5 a1i ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → i ∈ ℂ )
7 recn ⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℂ )
8 7 adantl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℂ )
9 6 8 mulcld ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( i · 𝑦 ) ∈ ℂ )
10 2 4 9 adddid ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 0 · ( 𝑥 + ( i · 𝑦 ) ) ) = ( ( 0 · 𝑥 ) + ( 0 · ( i · 𝑦 ) ) ) )
11 remul02 ⊢ ( 𝑥 ∈ ℝ → ( 0 · 𝑥 ) = 0 )
12 11 adantr ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 0 · 𝑥 ) = 0 )
13 sn-0tie0 ⊢ ( 0 · i ) = 0
14 13 oveq1i ⊢ ( ( 0 · i ) · 𝑦 ) = ( 0 · 𝑦 )
15 2 6 8 mulassd ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 0 · i ) · 𝑦 ) = ( 0 · ( i · 𝑦 ) ) )
16 remul02 ⊢ ( 𝑦 ∈ ℝ → ( 0 · 𝑦 ) = 0 )
17 16 adantl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 0 · 𝑦 ) = 0 )
18 14 15 17 3eqtr3a ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 0 · ( i · 𝑦 ) ) = 0 )
19 12 18 oveq12d ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 0 · 𝑥 ) + ( 0 · ( i · 𝑦 ) ) ) = ( 0 + 0 ) )
20 sn-00id ⊢ ( 0 + 0 ) = 0
21 19 20 eqtrdi ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 0 · 𝑥 ) + ( 0 · ( i · 𝑦 ) ) ) = 0 )
22 10 21 eqtrd ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 0 · ( 𝑥 + ( i · 𝑦 ) ) ) = 0 )
23 oveq2 ⊢ ( 𝐴 = ( 𝑥 + ( i · 𝑦 ) ) → ( 0 · 𝐴 ) = ( 0 · ( 𝑥 + ( i · 𝑦 ) ) ) )
24 23 eqeq1d ⊢ ( 𝐴 = ( 𝑥 + ( i · 𝑦 ) ) → ( ( 0 · 𝐴 ) = 0 ↔ ( 0 · ( 𝑥 + ( i · 𝑦 ) ) ) = 0 ) )
25 22 24 syl5ibrcom ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐴 = ( 𝑥 + ( i · 𝑦 ) ) → ( 0 · 𝐴 ) = 0 ) )
26 25 rexlimivv ⊢ ( ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝐴 = ( 𝑥 + ( i · 𝑦 ) ) → ( 0 · 𝐴 ) = 0 )
27 1 26 syl ⊢ ( 𝐴 ∈ ℂ → ( 0 · 𝐴 ) = 0 )