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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrrecval | Structured version Visualization version GIF version |
Description: The next step in the development of the surreals is to establish induction and recursion across left and right sets. To that end, we are going to develop a relationship 𝑅 that is founded, partial, and set-like across the surreals. This first theorem just establishes the value of 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
lrrec.1 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} |
Ref | Expression |
---|---|
lrrecval | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2839 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)) ↔ 𝐴 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))) | |
2 | fveq2 6663 | . . . 4 ⊢ (𝑦 = 𝐵 → ( L ‘𝑦) = ( L ‘𝐵)) | |
3 | fveq2 6663 | . . . 4 ⊢ (𝑦 = 𝐵 → ( R ‘𝑦) = ( R ‘𝐵)) | |
4 | 2, 3 | uneq12d 4071 | . . 3 ⊢ (𝑦 = 𝐵 → (( L ‘𝑦) ∪ ( R ‘𝑦)) = (( L ‘𝐵) ∪ ( R ‘𝐵))) |
5 | 4 | eleq2d 2837 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)) ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))) |
6 | lrrec.1 | . 2 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | |
7 | 1, 5, 6 | brabg 5400 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3858 class class class wbr 5036 {copab 5098 ‘cfv 6340 No csur 33440 L cleft 33623 R cright 33624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-v 3411 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-iota 6299 df-fv 6348 |
This theorem is referenced by: lrrecval2 33679 lrrecse 33681 lrrecpred 33683 |
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