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Mirrors > Home > ILE Home > Th. List > addcomi | GIF version |
Description: Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mul.1 | ⊢ 𝐴 ∈ ℂ |
mul.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
addcomi | ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mul.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | addcom 7867 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 + caddc 7591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia3 107 ax-addcom 7688 |
This theorem is referenced by: addcomli 7875 add42i 7896 mvlladdi 7948 3m1e2 8808 fztpval 9831 fzo0to42pr 9965 ef01bndlem 11390 tangtx 12846 |
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