ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addid0 GIF version

Theorem addid0 7443
Description: If adding a number to a another number yields the other number, the added number must be 0. This shows that 0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
Assertion
Ref Expression
addid0 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋 + 𝑌) = 𝑋𝑌 = 0))

Proof of Theorem addid0
StepHypRef Expression
1 simpl 106 . . . 4 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → 𝑋 ∈ ℂ)
2 simpr 107 . . . 4 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → 𝑌 ∈ ℂ)
31, 1, 2subaddd 7403 . . 3 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 𝑋))
4 eqcom 2058 . . . . 5 ((𝑋𝑋) = 𝑌𝑌 = (𝑋𝑋))
5 simpr 107 . . . . . . 7 ((𝑋 ∈ ℂ ∧ 𝑌 = (𝑋𝑋)) → 𝑌 = (𝑋𝑋))
6 subid 7293 . . . . . . . 8 (𝑋 ∈ ℂ → (𝑋𝑋) = 0)
76adantr 265 . . . . . . 7 ((𝑋 ∈ ℂ ∧ 𝑌 = (𝑋𝑋)) → (𝑋𝑋) = 0)
85, 7eqtrd 2088 . . . . . 6 ((𝑋 ∈ ℂ ∧ 𝑌 = (𝑋𝑋)) → 𝑌 = 0)
98ex 112 . . . . 5 (𝑋 ∈ ℂ → (𝑌 = (𝑋𝑋) → 𝑌 = 0))
104, 9syl5bi 145 . . . 4 (𝑋 ∈ ℂ → ((𝑋𝑋) = 𝑌𝑌 = 0))
1110adantr 265 . . 3 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋𝑋) = 𝑌𝑌 = 0))
123, 11sylbird 163 . 2 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋 + 𝑌) = 𝑋𝑌 = 0))
13 oveq2 5548 . . . . 5 (𝑌 = 0 → (𝑋 + 𝑌) = (𝑋 + 0))
14 addid1 7212 . . . . 5 (𝑋 ∈ ℂ → (𝑋 + 0) = 𝑋)
1513, 14sylan9eqr 2110 . . . 4 ((𝑋 ∈ ℂ ∧ 𝑌 = 0) → (𝑋 + 𝑌) = 𝑋)
1615ex 112 . . 3 (𝑋 ∈ ℂ → (𝑌 = 0 → (𝑋 + 𝑌) = 𝑋))
1716adantr 265 . 2 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → (𝑌 = 0 → (𝑋 + 𝑌) = 𝑋))
1812, 17impbid 124 1 ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋 + 𝑌) = 𝑋𝑌 = 0))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  (class class class)co 5540  cc 6945  0cc0 6947   + caddc 6950  cmin 7245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247
This theorem is referenced by:  addn0nid  7444
  Copyright terms: Public domain W3C validator