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Theorem pzriprnglem3 21494
Description: Lemma 3 for pzriprng 21508: An element of 𝐼 is an ordered pair. (Contributed by AV, 18-Mar-2025.)
Hypotheses
Ref Expression
pzriprng.r 𝑅 = (ℤring ×sring)
pzriprng.i 𝐼 = (ℤ × {0})
Assertion
Ref Expression
pzriprnglem3 (𝑋𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩)
Distinct variable group:   𝑥,𝑋
Allowed substitution hints:   𝑅(𝑥)   𝐼(𝑥)

Proof of Theorem pzriprnglem3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pzriprng.i . . 3 𝐼 = (ℤ × {0})
21eleq2i 2833 . 2 (𝑋𝐼𝑋 ∈ (ℤ × {0}))
3 elxp2 5709 . 2 (𝑋 ∈ (ℤ × {0}) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩)
4 0z 12624 . . . 4 0 ∈ ℤ
5 opeq2 4874 . . . . . 6 (𝑦 = 0 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0⟩)
65eqeq2d 2748 . . . . 5 (𝑦 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩))
76rexsng 4676 . . . 4 (0 ∈ ℤ → (∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩))
84, 7ax-mp 5 . . 3 (∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩)
98rexbii 3094 . 2 (∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩)
102, 3, 93bitri 297 1 (𝑋𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wrex 3070  {csn 4626  cop 4632   × cxp 5683  (class class class)co 7431  0cc0 11155  cz 12613   ×s cxps 17551  ringczring 21457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-1cn 11213  ax-addrcl 11216  ax-rnegex 11226  ax-cnre 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-iota 6514  df-fv 6569  df-ov 7434  df-neg 11495  df-z 12614
This theorem is referenced by:  pzriprnglem4  21495  pzriprnglem5  21496  pzriprnglem6  21497  pzriprnglem8  21499
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