Theorem xnn0xadd0 10101

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 9466	. . . 4 ⊢ (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0 ∨ 𝐴 = +∞))
2		elxnn0 9466	. . . . . . 7 ⊢ (𝐵 ∈ ℕ0* ↔ (𝐵 ∈ ℕ0 ∨ 𝐵 = +∞))
3		nn0re 9410	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
4		nn0re 9410	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ)
5		rexadd 10086	. . . . . . . . . . . . 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 2240	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 + 𝐵) = 0))
8		nn0ge0 9426	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
9	3, 8	jca 306	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
10		nn0ge0 9426	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ0 → 0 ≤ 𝐵)
11	4, 10	jca 306	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ0 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
12		add20 8653	. . . . . . . . . . . 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 6025	. . . . . . . . . . . . 13 ⊢ (𝐵 = +∞ → (𝐴 +𝑒 𝐵) = (𝐴 +𝑒 +∞))
18	17	eqeq1d 2240	. . . . . . . . . . . 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 9468	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℕ0*)
21		xnn0xrnemnf 9476	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞))
22		xaddpnf1 10080	. . . . . . . . . . . . . 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 2240	. . . . . . . . . . 11 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 +∞) = 0 ↔ +∞ = 0))
26	19, 25	bitrd 188	. . . . . . . . . 10 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0))
27		0re 8178	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
28		renepnf 8226	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
29	27, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ 0 ≠ +∞
30	29	nesymi 2448	. . . . . . . . . . 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 723	. . . . . . 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 6024	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 𝐵) = (+∞ +𝑒 𝐵))
38	37	eqeq1d 2240	. . . . . . . 8 ⊢ (𝐴 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (+∞ +𝑒 𝐵) = 0))
39		xnn0xrnemnf 9476	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ0* → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞))
40		xaddpnf2 10081	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (+∞ +𝑒 𝐵) = +∞)
41	39, 40	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ0* → (+∞ +𝑒 𝐵) = +∞)
42	41	eqeq1d 2240	. . . . . . . 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 723	. . . 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 6026	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = (0 +𝑒 0))
50		0xr 8225	. . . 4 ⊢ 0 ∈ ℝ*
51		xaddid1 10096	. . . 4 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
52	50, 51	ax-mp 5	. . 3 ⊢ (0 +𝑒 0) = 0
53	49, 52	eqtrdi 2280	. 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 715   = wceq 1397   ∈ wcel 2202   ≠ wne 2402   class class class wbr 4088  (class class class)co 6017  ℝcr 8030  0cc0 8031   + caddc 8034  +∞cpnf 8210  -∞cmnf 8211  ℝ^*cxr 8212   ≤ cle 8214  ℕ₀cn0 9401  ℕ₀^*cxnn0 9464   +_𝑒 cxad 10004

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-inn 9143  df-n0 9402  df-xnn0 9465  df-xadd 10007

This theorem is referenced by: (None)