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| Mirrors > Home > MPE Home > Th. List > pzriprnglem3 | Structured version Visualization version GIF version | ||
| Description: Lemma 3 for pzriprng 21508: 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 2833 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ (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〉) | |
| 6 | 5 | eqeq2d 2748 | . . . . 5 ⊢ (𝑦 = 0 → (𝑋 = 〈𝑥, 𝑦〉 ↔ 𝑋 = 〈𝑥, 0〉)) |
| 7 | 6 | rexsng 4676 | . . . 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 3094 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 ∈ ℤ 𝑋 = 〈𝑥, 0〉) |
| 10 | 2, 3, 9 | 3bitri 297 | 1 ⊢ (𝑋 ∈ 𝐼 ↔ ∃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 ℤcz 12613 ×s cxps 17551 ℤringczring 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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