Theorem xnn0xadd0 10200

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 9565	. . . 4 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
2		elxnn0 9565	. . . . . . 7 ⊢ (𝐵 ∈ ℕ0* ↔ (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞))
3		nn0re 9505	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
4		nn0re 9505	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ)
5		rexadd 10185	. . . . . . . . . . . . 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 2241	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 + 𝐵) = 0))
8		nn0ge0 9521	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
9	3, 8	jca 306	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
10		nn0ge0 9521	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ0 → 0 ≤ 𝐵)
11	4, 10	jca 306	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ0 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
12		add20 8748	. . . . . . . . . . . 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 6058	. . . . . . . . . . . . 13 ⊢ (𝐵 = +∞ → (𝐴 +𝑒 𝐵) = (𝐴 +𝑒 +∞))
18	17	eqeq1d 2241	. . . . . . . . . . . 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 9567	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
21		xnn0xrnemnf 9575	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
22		xaddpnf1 10179	. . . . . . . . . . . . . 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 2241	. . . . . . . . . . 11 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 +∞) = 0 ↔ +∞ = 0))
26	19, 25	bitrd 188	. . . . . . . . . 10 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0))
27		0re 8274	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
28		renepnf 8321	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ 0 ≠ +∞
30	29	nesymi 2458	. . . . . . . . . . 11 ⊢ ¬ +∞ = 0
31	30	pm2.21i 651	. . . . . . . . . 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 724	. . . . . . 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 6057	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 𝐵) = (+∞ +𝑒 𝐵))
38	37	eqeq1d 2241	. . . . . . . 8 ⊢ (𝐴 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (+∞ +𝑒 𝐵) = 0))
39		xnn0xrnemnf 9575	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ0* → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞))
40		xaddpnf2 10180	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (+∞ +𝑒 𝐵) = +∞)
41	39, 40	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ0* → (+∞ +𝑒 𝐵) = +∞)
42	41	eqeq1d 2241	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ0* → ((+∞ +𝑒 𝐵) = 0 ↔ +∞ = 0))
43	38, 42	sylan9bb 462	. . . . . . 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 724	. . . 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 6059	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = (0 +𝑒 0))
50		0xr 8320	. . . 4 ⊢ 0 ∈ ℝ*
51		xaddid1 10195	. . . 4 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
52	50, 51	ax-mp 5	. . 3 ⊢ (0 +𝑒 0) = 0
53	49, 52	eqtrdi 2281	. 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)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2203   ≠ wne 2412   class class class wbr 4109  (class class class)co 6050  ℝcr 8126  0cc0 8127   + caddc 8130  +∞cpnf 8305  -∞cmnf 8306  ℝ^*cxr 8307   ≤ cle 8309  ℕ₀cn0 9496  ℕ₀^*cxnn0 9563   +_𝑒 cxad 10103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-inn 9238  df-n0 9497  df-xnn0 9564  df-xadd 10106

This theorem is referenced by: (None)