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Mirrors > Home > MPE Home > Th. List > pzriprnglem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for pzriprng 21427: An element of 𝐼 is an ordered pair. (Contributed by AV, 18-Mar-2025.) |
Ref | Expression |
---|---|
pzriprng.r | ⊢ 𝑅 = (ℤring ×s ℤring) |
pzriprng.i | ⊢ 𝐼 = (ℤ × {0}) |
Ref | Expression |
---|---|
pzriprnglem3 | ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pzriprng.i | . . 3 ⊢ 𝐼 = (ℤ × {0}) | |
2 | 1 | eleq2i 2817 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ (ℤ × {0})) |
3 | elxp2 5696 | . 2 ⊢ (𝑋 ∈ (ℤ × {0}) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩) | |
4 | 0z 12599 | . . . 4 ⊢ 0 ∈ ℤ | |
5 | opeq2 4870 | . . . . . 6 ⊢ (𝑦 = 0 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0⟩) | |
6 | 5 | eqeq2d 2736 | . . . . 5 ⊢ (𝑦 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩)) |
7 | 6 | rexsng 4674 | . . . 4 ⊢ (0 ∈ ℤ → (∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩)) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ (∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, 0⟩) |
9 | 8 | rexbii 3084 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩) |
10 | 2, 3, 9 | 3bitri 296 | 1 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 {csn 4624 ⟨cop 4630 × cxp 5670 (class class class)co 7416 0cc0 11138 ℤcz 12588 ×s cxps 17487 ℤringczring 21376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-1cn 11196 ax-addrcl 11199 ax-rnegex 11209 ax-cnre 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-iota 6495 df-fv 6551 df-ov 7419 df-neg 11477 df-z 12589 |
This theorem is referenced by: pzriprnglem4 21414 pzriprnglem5 21415 pzriprnglem6 21416 pzriprnglem8 21418 |
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