![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cutpos | Structured version Visualization version GIF version |
Description: Reduce the elements of a cut for a positive number. (Contributed by Scott Fenton, 13-Mar-2025.) |
Ref | Expression |
---|---|
cutpos.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
cutpos.2 | ⊢ (𝜑 → 0s <s 𝐴) |
Ref | Expression |
---|---|
cutpos | ⊢ (𝜑 → 𝐴 = (({ 0s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) |s ( R ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lltropt 27812 | . . 3 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴)) |
3 | cutpos.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ No ) | |
4 | lrcut 27842 | . . . 4 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴) |
6 | 5 | eqcomd 2734 | . 2 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴))) |
7 | cutpos.2 | . . 3 ⊢ (𝜑 → 0s <s 𝐴) | |
8 | 3, 7 | 0elleft 27849 | . 2 ⊢ (𝜑 → 0s ∈ ( L ‘𝐴)) |
9 | 2, 6, 8 | cutlt 27865 | 1 ⊢ (𝜑 → 𝐴 = (({ 0s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) |s ( R ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3429 ∪ cun 3945 {csn 4629 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 No csur 27586 <s cslt 27587 <<s csslt 27726 |s cscut 27728 0s c0s 27768 L cleft 27785 R cright 27786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-1o 8487 df-2o 8488 df-no 27589 df-slt 27590 df-bday 27591 df-sle 27691 df-sslt 27727 df-scut 27729 df-0s 27770 df-made 27787 df-old 27788 df-left 27790 df-right 27791 |
This theorem is referenced by: precsexlem11 28128 |
Copyright terms: Public domain | W3C validator |