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Mirrors > Home > MPE Home > Th. List > cofcutrtime2d | Structured version Visualization version GIF version |
Description: If ð is a timely cut of ðī and ðĩ, then ( R âð) is coinitial with ðĩ. (Contributed by Scott Fenton, 23-Jan-2025.) |
Ref | Expression |
---|---|
cofcutrtimed.1 | âĒ (ð â (ðī ⊠ðĩ) â ( O â( bday âð))) |
cofcutrtimed.2 | âĒ (ð â ðī <<s ðĩ) |
cofcutrtimed.3 | âĒ (ð â ð = (ðī |s ðĩ)) |
Ref | Expression |
---|---|
cofcutrtime2d | âĒ (ð â âð§ â ðĩ âðĪ â ( R âð)ðĪ âĪs ð§) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofcutrtimed.1 | . . 3 âĒ (ð â (ðī ⊠ðĩ) â ( O â( bday âð))) | |
2 | cofcutrtimed.2 | . . 3 âĒ (ð â ðī <<s ðĩ) | |
3 | cofcutrtimed.3 | . . 3 âĒ (ð â ð = (ðī |s ðĩ)) | |
4 | cofcutrtime 27821 | . . 3 âĒ (((ðī ⊠ðĩ) â ( O â( bday âð)) â§ ðī <<s ðĩ â§ ð = (ðī |s ðĩ)) â (âðĨ â ðī âðĶ â ( L âð)ðĨ âĪs ðĶ â§ âð§ â ðĩ âðĪ â ( R âð)ðĪ âĪs ð§)) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . 2 âĒ (ð â (âðĨ â ðī âðĶ â ( L âð)ðĨ âĪs ðĶ â§ âð§ â ðĩ âðĪ â ( R âð)ðĪ âĪs ð§)) |
6 | 5 | simprd 495 | 1 âĒ (ð â âð§ â ðĩ âðĪ â ( R âð)ðĪ âĪs ð§) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 395 = wceq 1534 âwral 3056 âwrex 3065 ⊠cun 3942 â wss 3944 class class class wbr 5142 âcfv 6542 (class class class)co 7414 bday cbday 27549 âĪs csle 27651 <<s csslt 27687 |s cscut 27689 O cold 27744 L cleft 27746 R cright 27747 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-1o 8478 df-2o 8479 df-no 27550 df-slt 27551 df-bday 27552 df-sle 27652 df-sslt 27688 df-scut 27690 df-made 27748 df-old 27749 df-left 27751 df-right 27752 |
This theorem is referenced by: (None) |
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