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Theorem xnn0xadd0 9673
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
StepHypRef Expression
1 elxnn0 9061 . . . 4 (𝐴 ∈ ℕ0* ↔ (𝐴 ∈ ℕ0𝐴 = +∞))
2 elxnn0 9061 . . . . . . 7 (𝐵 ∈ ℕ0* ↔ (𝐵 ∈ ℕ0𝐵 = +∞))
3 nn0re 9005 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
4 nn0re 9005 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ0𝐵 ∈ ℝ)
5 rexadd 9658 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
63, 4, 5syl2an 287 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
76eqeq1d 2148 . . . . . . . . . . 11 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 + 𝐵) = 0))
8 nn0ge0 9021 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
93, 8jca 304 . . . . . . . . . . . 12 (𝐴 ∈ ℕ0 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
10 nn0ge0 9021 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ0 → 0 ≤ 𝐵)
114, 10jca 304 . . . . . . . . . . . 12 (𝐵 ∈ ℕ0 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
12 add20 8255 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
139, 11, 12syl2an 287 . . . . . . . . . . 11 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
147, 13bitrd 187 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
1514biimpd 143 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
1615expcom 115 . . . . . . . 8 (𝐵 ∈ ℕ0 → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
17 oveq2 5785 . . . . . . . . . . . . 13 (𝐵 = +∞ → (𝐴 +𝑒 𝐵) = (𝐴 +𝑒 +∞))
1817eqeq1d 2148 . . . . . . . . . . . 12 (𝐵 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 +𝑒 +∞) = 0))
1918adantr 274 . . . . . . . . . . 11 ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 +𝑒 +∞) = 0))
20 nn0xnn0 9063 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ0𝐴 ∈ ℕ0*)
21 xnn0xrnemnf 9071 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
22 xaddpnf1 9652 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
2320, 21, 223syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ0 → (𝐴 +𝑒 +∞) = +∞)
2423adantl 275 . . . . . . . . . . . 12 ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → (𝐴 +𝑒 +∞) = +∞)
2524eqeq1d 2148 . . . . . . . . . . 11 ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 +∞) = 0 ↔ +∞ = 0))
2619, 25bitrd 187 . . . . . . . . . 10 ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0))
27 0re 7785 . . . . . . . . . . . . 13 0 ∈ ℝ
28 renepnf 7832 . . . . . . . . . . . . 13 (0 ∈ ℝ → 0 ≠ +∞)
2927, 28ax-mp 5 . . . . . . . . . . . 12 0 ≠ +∞
3029nesymi 2354 . . . . . . . . . . 11 ¬ +∞ = 0
3130pm2.21i 635 . . . . . . . . . 10 (+∞ = 0 → (𝐴 = 0 ∧ 𝐵 = 0))
3226, 31syl6bi 162 . . . . . . . . 9 ((𝐵 = +∞ ∧ 𝐴 ∈ ℕ0) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
3332ex 114 . . . . . . . 8 (𝐵 = +∞ → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
3416, 33jaoi 705 . . . . . . 7 ((𝐵 ∈ ℕ0𝐵 = +∞) → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
352, 34sylbi 120 . . . . . 6 (𝐵 ∈ ℕ0* → (𝐴 ∈ ℕ0 → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
3635com12 30 . . . . 5 (𝐴 ∈ ℕ0 → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
37 oveq1 5784 . . . . . . . . 9 (𝐴 = +∞ → (𝐴 +𝑒 𝐵) = (+∞ +𝑒 𝐵))
3837eqeq1d 2148 . . . . . . . 8 (𝐴 = +∞ → ((𝐴 +𝑒 𝐵) = 0 ↔ (+∞ +𝑒 𝐵) = 0))
39 xnn0xrnemnf 9071 . . . . . . . . . 10 (𝐵 ∈ ℕ0* → (𝐵 ∈ ℝ*𝐵 ≠ -∞))
40 xaddpnf2 9653 . . . . . . . . . 10 ((𝐵 ∈ ℝ*𝐵 ≠ -∞) → (+∞ +𝑒 𝐵) = +∞)
4139, 40syl 14 . . . . . . . . 9 (𝐵 ∈ ℕ0* → (+∞ +𝑒 𝐵) = +∞)
4241eqeq1d 2148 . . . . . . . 8 (𝐵 ∈ ℕ0* → ((+∞ +𝑒 𝐵) = 0 ↔ +∞ = 0))
4338, 42sylan9bb 457 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ +∞ = 0))
4443, 31syl6bi 162 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
4544ex 114 . . . . 5 (𝐴 = +∞ → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
4636, 45jaoi 705 . . . 4 ((𝐴 ∈ ℕ0𝐴 = +∞) → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
471, 46sylbi 120 . . 3 (𝐴 ∈ ℕ0* → (𝐵 ∈ ℕ0* → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0))))
4847imp 123 . 2 ((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 → (𝐴 = 0 ∧ 𝐵 = 0)))
49 oveq12 5786 . . 3 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = (0 +𝑒 0))
50 0xr 7831 . . . 4 0 ∈ ℝ*
51 xaddid1 9668 . . . 4 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
5250, 51ax-mp 5 . . 3 (0 +𝑒 0) = 0
5349, 52eqtrdi 2188 . 2 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 +𝑒 𝐵) = 0)
5448, 53impbid1 141 1 ((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 697   = wceq 1331  wcel 1480  wne 2308   class class class wbr 3932  (class class class)co 5777  cr 7638  0cc0 7639   + caddc 7642  +∞cpnf 7816  -∞cmnf 7817  *cxr 7818  cle 7820  0cn0 8996  0*cxnn0 9059   +𝑒 cxad 9580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-addcom 7739  ax-addass 7741  ax-i2m1 7744  ax-0lt1 7745  ax-0id 7747  ax-rnegex 7748  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-br 3933  df-opab 3993  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-iota 5091  df-fun 5128  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-inn 8740  df-n0 8997  df-xnn0 9060  df-xadd 9583
This theorem is referenced by: (None)
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