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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrneq3 | Structured version Visualization version GIF version |
Description: Two translations agree at any atom not under the fiducial co-atom π iff they are equal. (Contributed by NM, 25-Jul-2013.) |
Ref | Expression |
---|---|
cdlemd.l | β’ β€ = (leβπΎ) |
cdlemd.a | β’ π΄ = (AtomsβπΎ) |
cdlemd.h | β’ π» = (LHypβπΎ) |
cdlemd.t | β’ π = ((LTrnβπΎ)βπ) |
Ref | Expression |
---|---|
ltrneq3 | β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉβπ) = (πΊβπ) β πΉ = πΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1188 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉβπ) = (πΊβπ)) β (πΎ β HL β§ π β π»)) | |
2 | simpl2l 1223 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉβπ) = (πΊβπ)) β πΉ β π) | |
3 | simpl2r 1224 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉβπ) = (πΊβπ)) β πΊ β π) | |
4 | simpl3 1190 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉβπ) = (πΊβπ)) β (π β π΄ β§ Β¬ π β€ π)) | |
5 | simpr 484 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉβπ) = (πΊβπ)) β (πΉβπ) = (πΊβπ)) | |
6 | cdlemd.l | . . . 4 β’ β€ = (leβπΎ) | |
7 | cdlemd.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
8 | cdlemd.h | . . . 4 β’ π» = (LHypβπΎ) | |
9 | cdlemd.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
10 | 6, 7, 8, 9 | cdlemd 39590 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (πΉβπ) = (πΊβπ)) β πΉ = πΊ) |
11 | 1, 2, 3, 4, 5, 10 | syl311anc 1381 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (πΉβπ) = (πΊβπ)) β πΉ = πΊ) |
12 | fveq1 6883 | . . 3 β’ (πΉ = πΊ β (πΉβπ) = (πΊβπ)) | |
13 | 12 | adantl 481 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ πΉ = πΊ) β (πΉβπ) = (πΊβπ)) |
14 | 11, 13 | impbida 798 | 1 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉβπ) = (πΊβπ) β πΉ = πΊ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 lecple 17210 Atomscatm 38645 HLchlt 38732 LHypclh 39367 LTrncltrn 39484 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 |
This theorem is referenced by: cdlemn3 40580 |
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