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Mirrors > Home > MPE Home > Th. List > cofcut1d | Structured version Visualization version GIF version |
Description: If ðķ is cofinal with ðī and ð· is coinitial with ðĩ and the cut of ðī and ðĩ lies between ðķ and ð·, then the cut of ðķ and ð· is equal to the cut of ðī and ðĩ. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.) |
Ref | Expression |
---|---|
cofcut1d.1 | âĒ (ð â ðī <<s ðĩ) |
cofcut1d.2 | âĒ (ð â âðĨ â ðī âðĶ â ðķ ðĨ âĪs ðĶ) |
cofcut1d.3 | âĒ (ð â âð§ â ðĩ âðĪ â ð· ðĪ âĪs ð§) |
cofcut1d.4 | âĒ (ð â ðķ <<s {(ðī |s ðĩ)}) |
cofcut1d.5 | âĒ (ð â {(ðī |s ðĩ)} <<s ð·) |
Ref | Expression |
---|---|
cofcut1d | âĒ (ð â (ðī |s ðĩ) = (ðķ |s ð·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofcut1d.1 | . 2 âĒ (ð â ðī <<s ðĩ) | |
2 | cofcut1d.2 | . 2 âĒ (ð â âðĨ â ðī âðĶ â ðķ ðĨ âĪs ðĶ) | |
3 | cofcut1d.3 | . 2 âĒ (ð â âð§ â ðĩ âðĪ â ð· ðĪ âĪs ð§) | |
4 | cofcut1d.4 | . 2 âĒ (ð â ðķ <<s {(ðī |s ðĩ)}) | |
5 | cofcut1d.5 | . 2 âĒ (ð â {(ðī |s ðĩ)} <<s ð·) | |
6 | cofcut1 27744 | . 2 âĒ ((ðī <<s ðĩ â§ (âðĨ â ðī âðĶ â ðķ ðĨ âĪs ðĶ â§ âð§ â ðĩ âðĪ â ð· ðĪ âĪs ð§) â§ (ðķ <<s {(ðī |s ðĩ)} â§ {(ðī |s ðĩ)} <<s ð·)) â (ðī |s ðĩ) = (ðķ |s ð·)) | |
7 | 1, 2, 3, 4, 5, 6 | syl122anc 1376 | 1 âĒ (ð â (ðī |s ðĩ) = (ðķ |s ð·)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 = wceq 1533 âwral 3053 âwrex 3062 {csn 4620 class class class wbr 5138 (class class class)co 7401 âĪs csle 27581 <<s csslt 27617 |s cscut 27619 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1o 8461 df-2o 8462 df-no 27480 df-slt 27481 df-bday 27482 df-sle 27582 df-sslt 27618 df-scut 27620 |
This theorem is referenced by: addsuniflem 27822 negsunif 27871 mulsuniflem 27953 |
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