Theorem xnn0xadd0

Description: The sum of two extended nonnegative integers is 0 iff each of the two extended nonnegative integers is 0 . (Contributed by AV, 14-Dec-2020)

	Ref	Expression
Assertion	xnn0xadd0	⊢ ( ( 𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0* ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elxnn0	⊢ ( 𝐴 ∈ ℕ0* ↔ ( 𝐴 ∈ ℕ0 ∨ 𝐴 = +∞ ) )
2		elxnn0	⊢ ( 𝐵 ∈ ℕ0* ↔ ( 𝐵 ∈ ℕ0 ∨ 𝐵 = +∞ ) )
3		nn0re	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ )
4		nn0re	⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ )
5		rexadd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 +e 𝐵 ) = ( 𝐴 + 𝐵 ) )
6	3 4 5	syl2an	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 +e 𝐵 ) = ( 𝐴 + 𝐵 ) )
7	6	eqeq1d	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( 𝐴 + 𝐵 ) = 0 ) )
8		nn0ge0	⊢ ( 𝐴 ∈ ℕ0 → 0 ≤ 𝐴 )
9	3 8	jca	⊢ ( 𝐴 ∈ ℕ0 → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
10		nn0ge0	⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ 𝐵 )
11	4 10	jca	⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) )
12		add20	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
13	9 11 12	syl2an	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
14	7 13	bitrd	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
15	14	biimpd	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
16	15	expcom	⊢ ( 𝐵 ∈ ℕ0 → ( 𝐴 ∈ ℕ0 → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
17		oveq2	⊢ ( 𝐵 = +∞ → ( 𝐴 +e 𝐵 ) = ( 𝐴 +e +∞ ) )
18	17	eqeq1d	⊢ ( 𝐵 = +∞ → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( 𝐴 +e +∞ ) = 0 ) )
19	18	adantr	⊢ ( ( 𝐵 = +∞ ∧ 𝐴 ∈ ℕ0 ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( 𝐴 +e +∞ ) = 0 ) )
20		nn0xnn0	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0* )
21		xnn0xrnemnf	⊢ ( 𝐴 ∈ ℕ0* → ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞ ) )
22		xaddpnf1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞ ) → ( 𝐴 +e +∞ ) = +∞ )
23	20 21 22	3syl	⊢ ( 𝐴 ∈ ℕ0 → ( 𝐴 +e +∞ ) = +∞ )
24	23	adantl	⊢ ( ( 𝐵 = +∞ ∧ 𝐴 ∈ ℕ0 ) → ( 𝐴 +e +∞ ) = +∞ )
25	24	eqeq1d	⊢ ( ( 𝐵 = +∞ ∧ 𝐴 ∈ ℕ0 ) → ( ( 𝐴 +e +∞ ) = 0 ↔ +∞ = 0 ) )
26	19 25	bitrd	⊢ ( ( 𝐵 = +∞ ∧ 𝐴 ∈ ℕ0 ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ +∞ = 0 ) )
27		0re	⊢ 0 ∈ ℝ
28		renepnf	⊢ ( 0 ∈ ℝ → 0 ≠ +∞ )
29	27 28	ax-mp	⊢ 0 ≠ +∞
30	29	nesymi	⊢ ¬ +∞ = 0
31	30	pm2.21i	⊢ ( +∞ = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) )
32	26 31	biimtrdi	⊢ ( ( 𝐵 = +∞ ∧ 𝐴 ∈ ℕ0 ) → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
33	32	ex	⊢ ( 𝐵 = +∞ → ( 𝐴 ∈ ℕ0 → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
34	16 33	jaoi	⊢ ( ( 𝐵 ∈ ℕ0 ∨ 𝐵 = +∞ ) → ( 𝐴 ∈ ℕ0 → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
35	2 34	sylbi	⊢ ( 𝐵 ∈ ℕ0* → ( 𝐴 ∈ ℕ0 → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
36	35	com12	⊢ ( 𝐴 ∈ ℕ0 → ( 𝐵 ∈ ℕ0* → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
37		oveq1	⊢ ( 𝐴 = +∞ → ( 𝐴 +e 𝐵 ) = ( +∞ +e 𝐵 ) )
38	37	eqeq1d	⊢ ( 𝐴 = +∞ → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( +∞ +e 𝐵 ) = 0 ) )
39		xnn0xrnemnf	⊢ ( 𝐵 ∈ ℕ0* → ( 𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞ ) )
40		xaddpnf2	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞ ) → ( +∞ +e 𝐵 ) = +∞ )
41	39 40	syl	⊢ ( 𝐵 ∈ ℕ0* → ( +∞ +e 𝐵 ) = +∞ )
42	41	eqeq1d	⊢ ( 𝐵 ∈ ℕ0* → ( ( +∞ +e 𝐵 ) = 0 ↔ +∞ = 0 ) )
43	38 42	sylan9bb	⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℕ0* ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ +∞ = 0 ) )
44	43 31	biimtrdi	⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℕ0* ) → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
45	44	ex	⊢ ( 𝐴 = +∞ → ( 𝐵 ∈ ℕ0* → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
46	36 45	jaoi	⊢ ( ( 𝐴 ∈ ℕ0 ∨ 𝐴 = +∞ ) → ( 𝐵 ∈ ℕ0* → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
47	1 46	sylbi	⊢ ( 𝐴 ∈ ℕ0* → ( 𝐵 ∈ ℕ0* → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) )
48	47	imp	⊢ ( ( 𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0* ) → ( ( 𝐴 +e 𝐵 ) = 0 → ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
49		oveq12	⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( 𝐴 +e 𝐵 ) = ( 0 +e 0 ) )
50		0xr	⊢ 0 ∈ ℝ*
51		xaddrid	⊢ ( 0 ∈ ℝ* → ( 0 +e 0 ) = 0 )
52	50 51	ax-mp	⊢ ( 0 +e 0 ) = 0
53	49 52	eqtrdi	⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( 𝐴 +e 𝐵 ) = 0 )
54	48 53	impbid1	⊢ ( ( 𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0* ) → ( ( 𝐴 +e 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )