This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sn-addcan2d

Description: addcan2d without ax-mulcom . (Contributed by SN, 5-May-2024)

Ref Expression
Hypotheses sn-addcan2d.a ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
sn-addcan2d.b ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
sn-addcan2d.c ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion sn-addcan2d ( 𝜑 → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 sn-addcan2d.a ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 sn-addcan2d.b ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 sn-addcan2d.c ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 sn-negex ⊢ ( 𝐶 ∈ ℂ → ∃ 𝑥 ∈ ℂ ( 𝐶 + 𝑥 ) = 0 )
5 3 4 syl ⊢ ( 𝜑 → ∃ 𝑥 ∈ ℂ ( 𝐶 + 𝑥 ) = 0 )
6 oveq1 ⊢ ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) → ( ( 𝐴 + 𝐶 ) + 𝑥 ) = ( ( 𝐵 + 𝐶 ) + 𝑥 ) )
7 1 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → 𝐴 ∈ ℂ )
8 3 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → 𝐶 ∈ ℂ )
9 simprl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → 𝑥 ∈ ℂ )
10 7 8 9 addassd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( 𝐴 + 𝐶 ) + 𝑥 ) = ( 𝐴 + ( 𝐶 + 𝑥 ) ) )
11 simprr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( 𝐶 + 𝑥 ) = 0 )
12 11 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( 𝐴 + ( 𝐶 + 𝑥 ) ) = ( 𝐴 + 0 ) )
13 sn-addrid ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + 0 ) = 𝐴 )
14 7 13 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( 𝐴 + 0 ) = 𝐴 )
15 10 12 14 3eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( 𝐴 + 𝐶 ) + 𝑥 ) = 𝐴 )
16 2 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → 𝐵 ∈ ℂ )
17 16 8 9 addassd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( 𝐵 + 𝐶 ) + 𝑥 ) = ( 𝐵 + ( 𝐶 + 𝑥 ) ) )
18 11 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( 𝐵 + ( 𝐶 + 𝑥 ) ) = ( 𝐵 + 0 ) )
19 sn-addrid ⊢ ( 𝐵 ∈ ℂ → ( 𝐵 + 0 ) = 𝐵 )
20 16 19 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( 𝐵 + 0 ) = 𝐵 )
21 17 18 20 3eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( 𝐵 + 𝐶 ) + 𝑥 ) = 𝐵 )
22 15 21 eqeq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( ( 𝐴 + 𝐶 ) + 𝑥 ) = ( ( 𝐵 + 𝐶 ) + 𝑥 ) ↔ 𝐴 = 𝐵 ) )
23 6 22 imbitrid ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) → 𝐴 = 𝐵 ) )
24 oveq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) )
25 23 24 impbid1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ ( 𝐶 + 𝑥 ) = 0 ) ) → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )
26 5 25 rexlimddv ⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )