Theorem xnn0xadd0 10269

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.)

Assertion

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

Proof of Theorem xnn0xadd0

Step	Hyp	Ref	Expression
1		elxnn0 9632	. . . 4 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
2		elxnn0 9632	. . . . . . 7 ⊢ (𝐵 ∈ ℕ0* ↔ (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞))
3		nn0re 9572	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
4		nn0re 9572	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ)
5		rexadd 10254	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
6	3, 4, 5	syl2an 289	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
7	6	eqeq1d 2247	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 + 𝐵) = 0))
8		nn0ge0 9588	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
9	3, 8	jca 306	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
10		nn0ge0 9588	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ0 → 0 ≤ 𝐵)
11	4, 10	jca 306	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ0 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
12		add20 8802	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
13	9, 11, 12	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
14	7, 13	bitrd 188	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
15	14	biimpd 144	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
16	15	expcom 116	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ0 → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
17		oveq2 6093	. . . . . . . . . . . . 13 ⊢ (𝐵 = +∞ → (𝐴 +𝑒 𝐵) = (𝐴 +𝑒 +∞))
18	17	eqeq1d 2247	. . . . . . . . . . . 12 ⊢ (𝐵 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 +𝑒 +∞) = 0))
19	18	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 +𝑒 +∞) = 0))
20		nn0xnn0 9634	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
21		xnn0xrnemnf 9642	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
22		xaddpnf1 10248	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
23	20, 21, 22	3syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → (𝐴 +𝑒 +∞) = +∞)
24	23	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → (𝐴 +𝑒 +∞) = +∞)
25	24	eqeq1d 2247	. . . . . . . . . . 11 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 +∞) = 0 ↔ +∞ = 0))
26	19, 25	bitrd 188	. . . . . . . . . 10 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0))
27		0re 8326	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
28		renepnf 8373	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ 0 ≠ +∞
30	29	nesymi 2466	. . . . . . . . . . 11 ⊢ ¬ +∞ = 0
31	30	pm2.21i 655	. . . . . . . . . 10 ⊢ (+∞ = 0 → (𝐴 = 0 ∧ 𝐵 = 0))
32	26, 31	biimtrdi 163	. . . . . . . . 9 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
33	32	ex 115	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
34	16, 33	jaoi 728	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ0 ∨ 𝐵 = +∞) → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
35	2, 34	sylbi 121	. . . . . 6 ⊢ (𝐵 ∈ ℕ0* → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
36	35	com12 30	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
37		oveq1 6092	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 𝐵) = (+∞ +𝑒 𝐵))
38	37	eqeq1d 2247	. . . . . . . 8 ⊢ (𝐴 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (+∞ +𝑒 𝐵) = 0))
39		xnn0xrnemnf 9642	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ0* → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞))
40		xaddpnf2 10249	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (+∞ +𝑒 𝐵) = +∞)
41	39, 40	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ0* → (+∞ +𝑒 𝐵) = +∞)
42	41	eqeq1d 2247	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ0* → ((+∞ +𝑒 𝐵) = 0 ↔ +∞ = 0))
43	38, 42	sylan9bb 466	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0))
44	43, 31	biimtrdi 163	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
45	44	ex 115	. . . . 5 ⊢ (𝐴 = +∞ → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
46	36, 45	jaoi 728	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∨ 𝐴 = +∞) → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
47	1, 46	sylbi 121	. . 3 ⊢ (𝐴 ∈ ℕ0* → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
48	47	imp 124	. 2 ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
49		oveq12 6094	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = (0 +𝑒 0))
50		0xr 8372	. . . 4 ⊢ 0 ∈ ℝ*
51		xaddid1 10264	. . . 4 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
52	50, 51	ax-mp 5	. . 3 ⊢ (0 +𝑒 0) = 0
53	49, 52	eqtrdi 2287	. 2 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = 0)
54	48, 53	impbid1 142	1 ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   class class class wbr 4130  (class class class)co 6085  ℝcr 8178  0cc0 8179   + caddc 8182  +∞cpnf 8357  -∞cmnf 8358  ℝ^*cxr 8359   ≤ cle 8361  ℕ₀cn0 9563  ℕ₀^*cxnn0 9630   +_𝑒 cxad 10172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-inn 9305  df-n0 9564  df-xnn0 9631  df-xadd 10175

This theorem is used by: (None)