|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > cofcut2d | Structured version Visualization version GIF version | ||
| Description: If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| cofcut2d.1 | ⊢ (𝜑 → 𝐴 <<s 𝐵) | 
| cofcut2d.2 | ⊢ (𝜑 → 𝐶 ∈ 𝒫 No ) | 
| cofcut2d.3 | ⊢ (𝜑 → 𝐷 ∈ 𝒫 No ) | 
| cofcut2d.4 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) | 
| cofcut2d.5 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) | 
| cofcut2d.6 | ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢) | 
| cofcut2d.7 | ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟) | 
| Ref | Expression | 
|---|---|
| cofcut2d | ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cofcut2d.1 | . 2 ⊢ (𝜑 → 𝐴 <<s 𝐵) | |
| 2 | cofcut2d.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝒫 No ) | |
| 3 | cofcut2d.3 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝒫 No ) | |
| 4 | cofcut2d.4 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) | |
| 5 | cofcut2d.5 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) | |
| 6 | cofcut2d.6 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢) | |
| 7 | cofcut2d.7 | . 2 ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟) | |
| 8 | cofcut2 27957 | . 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | syl322anc 1399 | 1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 𝒫 cpw 4599 class class class wbr 5142 (class class class)co 7432 No csur 27685 ≤s csle 27790 <<s csslt 27826 |s cscut 27828 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1o 8507 df-2o 8508 df-no 27688 df-slt 27689 df-bday 27690 df-sle 27791 df-sslt 27827 df-scut 27829 | 
| This theorem is referenced by: cutlt 27967 | 
| Copyright terms: Public domain | W3C validator |