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Theorem redc0 13590
 Description: Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
Assertion
Ref Expression
redc0 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem redc0
StepHypRef Expression
1 0re 7861 . . . . 5 0 ∈ ℝ
2 eqeq1 2164 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
32dcbid 824 . . . . . 6 (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦DECID 𝑧 = 𝑦))
4 eqeq2 2167 . . . . . . 7 (𝑦 = 0 → (𝑧 = 𝑦𝑧 = 0))
54dcbid 824 . . . . . 6 (𝑦 = 0 → (DECID 𝑧 = 𝑦DECID 𝑧 = 0))
63, 5rspc2v 2829 . . . . 5 ((𝑧 ∈ ℝ ∧ 0 ∈ ℝ) → (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦DECID 𝑧 = 0))
71, 6mpan2 422 . . . 4 (𝑧 ∈ ℝ → (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦DECID 𝑧 = 0))
87impcom 124 . . 3 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦𝑧 ∈ ℝ) → DECID 𝑧 = 0)
98ralrimiva 2530 . 2 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
10 eqeq1 2164 . . . . . 6 (𝑧 = (𝑥𝑦) → (𝑧 = 0 ↔ (𝑥𝑦) = 0))
1110dcbid 824 . . . . 5 (𝑧 = (𝑥𝑦) → (DECID 𝑧 = 0 ↔ DECID (𝑥𝑦) = 0))
12 simpl 108 . . . . 5 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
13 resubcl 8122 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥𝑦) ∈ ℝ)
1413adantl 275 . . . . 5 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥𝑦) ∈ ℝ)
1511, 12, 14rspcdva 2821 . . . 4 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → DECID (𝑥𝑦) = 0)
16 simprl 521 . . . . . . 7 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
1716recnd 7889 . . . . . 6 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℂ)
18 simprr 522 . . . . . . 7 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
1918recnd 7889 . . . . . 6 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℂ)
2017, 19subeq0ad 8179 . . . . 5 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥𝑦) = 0 ↔ 𝑥 = 𝑦))
2120dcbid 824 . . . 4 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (DECID (𝑥𝑦) = 0 ↔ DECID 𝑥 = 𝑦))
2215, 21mpbid 146 . . 3 ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → DECID 𝑥 = 𝑦)
2322ralrimivva 2539 . 2 (∀𝑧 ∈ ℝ DECID 𝑧 = 0 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
249, 23impbii 125 1 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 820   = wceq 1335   ∈ wcel 2128  ∀wral 2435  (class class class)co 5818  ℝcr 7714  0cc0 7715   − cmin 8029 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-setind 4494  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-addcom 7815  ax-addass 7817  ax-distr 7819  ax-i2m1 7820  ax-0id 7823  ax-rnegex 7824  ax-cnre 7826 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-sub 8031  df-neg 8032 This theorem is referenced by:  dcapnconstALT  13594
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