Proof of Theorem xnn0xadd0
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elxnn0 12601 | . . . 4
⊢ (𝐴 ∈
ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞)) | 
| 2 |  | elxnn0 12601 | . . . . . . 7
⊢ (𝐵 ∈
ℕ0* ↔ (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞)) | 
| 3 |  | nn0re 12535 | . . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℝ) | 
| 4 |  | nn0re 12535 | . . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℕ0
→ 𝐵 ∈
ℝ) | 
| 5 |  | rexadd 13274 | . . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | 
| 6 | 3, 4, 5 | syl2an 596 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | 
| 7 | 6 | eqeq1d 2739 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 + 𝐵) = 0)) | 
| 8 |  | nn0ge0 12551 | . . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℕ0
→ 0 ≤ 𝐴) | 
| 9 | 3, 8 | jca 511 | . . . . . . . . . . . 12
⊢ (𝐴 ∈ ℕ0
→ (𝐴 ∈ ℝ
∧ 0 ≤ 𝐴)) | 
| 10 |  | nn0ge0 12551 | . . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℕ0
→ 0 ≤ 𝐵) | 
| 11 | 4, 10 | jca 511 | . . . . . . . . . . . 12
⊢ (𝐵 ∈ ℕ0
→ (𝐵 ∈ ℝ
∧ 0 ≤ 𝐵)) | 
| 12 |  | add20 11775 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 13 | 9, 11, 12 | syl2an 596 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 14 | 7, 13 | bitrd 279 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 15 | 14 | biimpd 229 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 16 | 15 | expcom 413 | . . . . . . . 8
⊢ (𝐵 ∈ ℕ0
→ (𝐴 ∈
ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 17 |  | oveq2 7439 | . . . . . . . . . . . . 13
⊢ (𝐵 = +∞ → (𝐴 +𝑒 𝐵) = (𝐴 +𝑒
+∞)) | 
| 18 | 17 | eqeq1d 2739 | . . . . . . . . . . . 12
⊢ (𝐵 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 +𝑒 +∞) =
0)) | 
| 19 | 18 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0)
→ ((𝐴
+𝑒 𝐵) =
0 ↔ (𝐴
+𝑒 +∞) = 0)) | 
| 20 |  | nn0xnn0 12603 | . . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℕ0*) | 
| 21 |  | xnn0xrnemnf 12611 | . . . . . . . . . . . . . 14
⊢ (𝐴 ∈
ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠
-∞)) | 
| 22 |  | xaddpnf1 13268 | . . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ*
∧ 𝐴 ≠ -∞)
→ (𝐴
+𝑒 +∞) = +∞) | 
| 23 | 20, 21, 22 | 3syl 18 | . . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℕ0
→ (𝐴
+𝑒 +∞) = +∞) | 
| 24 | 23 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0)
→ (𝐴
+𝑒 +∞) = +∞) | 
| 25 | 24 | eqeq1d 2739 | . . . . . . . . . . 11
⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0)
→ ((𝐴
+𝑒 +∞) = 0 ↔ +∞ = 0)) | 
| 26 | 19, 25 | bitrd 279 | . . . . . . . . . 10
⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0)
→ ((𝐴
+𝑒 𝐵) =
0 ↔ +∞ = 0)) | 
| 27 |  | 0re 11263 | . . . . . . . . . . . . 13
⊢ 0 ∈
ℝ | 
| 28 |  | renepnf 11309 | . . . . . . . . . . . . 13
⊢ (0 ∈
ℝ → 0 ≠ +∞) | 
| 29 | 27, 28 | ax-mp 5 | . . . . . . . . . . . 12
⊢ 0 ≠
+∞ | 
| 30 | 29 | nesymi 2998 | . . . . . . . . . . 11
⊢  ¬
+∞ = 0 | 
| 31 | 30 | pm2.21i 119 | . . . . . . . . . 10
⊢ (+∞
= 0 → (𝐴 = 0 ∧
𝐵 = 0)) | 
| 32 | 26, 31 | biimtrdi 253 | . . . . . . . . 9
⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0)
→ ((𝐴
+𝑒 𝐵) =
0 → (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 33 | 32 | ex 412 | . . . . . . . 8
⊢ (𝐵 = +∞ → (𝐴 ∈ ℕ0
→ ((𝐴
+𝑒 𝐵) =
0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 34 | 16, 33 | jaoi 858 | . . . . . . 7
⊢ ((𝐵 ∈ ℕ0 ∨
𝐵 = +∞) → (𝐴 ∈ ℕ0
→ ((𝐴
+𝑒 𝐵) =
0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 35 | 2, 34 | sylbi 217 | . . . . . 6
⊢ (𝐵 ∈
ℕ0* → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 36 | 35 | com12 32 | . . . . 5
⊢ (𝐴 ∈ ℕ0
→ (𝐵 ∈
ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 37 |  | oveq1 7438 | . . . . . . . . 9
⊢ (𝐴 = +∞ → (𝐴 +𝑒 𝐵) = (+∞
+𝑒 𝐵)) | 
| 38 | 37 | eqeq1d 2739 | . . . . . . . 8
⊢ (𝐴 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (+∞
+𝑒 𝐵) =
0)) | 
| 39 |  | xnn0xrnemnf 12611 | . . . . . . . . . 10
⊢ (𝐵 ∈
ℕ0* → (𝐵 ∈ ℝ* ∧ 𝐵 ≠
-∞)) | 
| 40 |  | xaddpnf2 13269 | . . . . . . . . . 10
⊢ ((𝐵 ∈ ℝ*
∧ 𝐵 ≠ -∞)
→ (+∞ +𝑒 𝐵) = +∞) | 
| 41 | 39, 40 | syl 17 | . . . . . . . . 9
⊢ (𝐵 ∈
ℕ0* → (+∞ +𝑒 𝐵) = +∞) | 
| 42 | 41 | eqeq1d 2739 | . . . . . . . 8
⊢ (𝐵 ∈
ℕ0* → ((+∞ +𝑒 𝐵) = 0 ↔ +∞ =
0)) | 
| 43 | 38, 42 | sylan9bb 509 | . . . . . . 7
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈
ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0)) | 
| 44 | 43, 31 | biimtrdi 253 | . . . . . 6
⊢ ((𝐴 = +∞ ∧ 𝐵 ∈
ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 45 | 44 | ex 412 | . . . . 5
⊢ (𝐴 = +∞ → (𝐵 ∈
ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 46 | 36, 45 | jaoi 858 | . . . 4
⊢ ((𝐴 ∈ ℕ0 ∨
𝐴 = +∞) → (𝐵 ∈
ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 47 | 1, 46 | sylbi 217 | . . 3
⊢ (𝐴 ∈
ℕ0* → (𝐵 ∈ ℕ0*
→ ((𝐴
+𝑒 𝐵) =
0 → (𝐴 = 0 ∧ 𝐵 = 0)))) | 
| 48 | 47 | imp 406 | . 2
⊢ ((𝐴 ∈
ℕ0* ∧ 𝐵 ∈ ℕ0*)
→ ((𝐴
+𝑒 𝐵) =
0 → (𝐴 = 0 ∧ 𝐵 = 0))) | 
| 49 |  | oveq12 7440 | . . 3
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = (0 +𝑒
0)) | 
| 50 |  | 0xr 11308 | . . . 4
⊢ 0 ∈
ℝ* | 
| 51 |  | xaddrid 13283 | . . . 4
⊢ (0 ∈
ℝ* → (0 +𝑒 0) = 0) | 
| 52 | 50, 51 | ax-mp 5 | . . 3
⊢ (0
+𝑒 0) = 0 | 
| 53 | 49, 52 | eqtrdi 2793 | . 2
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = 0) | 
| 54 | 48, 53 | impbid1 225 | 1
⊢ ((𝐴 ∈
ℕ0* ∧ 𝐵 ∈ ℕ0*)
→ ((𝐴
+𝑒 𝐵) =
0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |