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Theorem pzriprnglem3 21511
Description: Lemma 3 for pzriprng 21525: 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 2830 . 2 (𝑋𝐼𝑋 ∈ (ℤ × {0}))
3 elxp2 5712 . 2 (𝑋 ∈ (ℤ × {0}) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩)
4 0z 12621 . . . 4 0 ∈ ℤ
5 opeq2 4878 . . . . . 6 (𝑦 = 0 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0⟩)
65eqeq2d 2745 . . . . 5 (𝑦 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩))
76rexsng 4680 . . . 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 3091 . 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 1536  wcel 2105  wrex 3067  {csn 4630  cop 4636   × cxp 5686  (class class class)co 7430  0cc0 11152  cz 12610   ×s cxps 17552  ringczring 21474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-1cn 11210  ax-addrcl 11213  ax-rnegex 11223  ax-cnre 11225
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-iota 6515  df-fv 6570  df-ov 7433  df-neg 11492  df-z 12611
This theorem is referenced by:  pzriprnglem4  21512  pzriprnglem5  21513  pzriprnglem6  21514  pzriprnglem8  21516
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