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| Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcocnvat | Structured version Visualization version GIF version | ||
| Description: Commonly used special case of trlcoat 39383. (Contributed by NM, 1-Jul-2013.) |
| Ref | Expression |
|---|---|
| trlcoat.a | β’ π΄ = (AtomsβπΎ) |
| trlcoat.h | β’ π» = (LHypβπΎ) |
| trlcoat.t | β’ π = ((LTrnβπΎ)βπ) |
| trlcoat.r | β’ π = ((trLβπΎ)βπ) |
| Ref | Expression |
|---|---|
| trlcocnvat | β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β (π β(πΉ β β‘πΊ)) β π΄) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1136 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β (πΎ β HL β§ π β π»)) | |
| 2 | simp2l 1199 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β πΉ β π) | |
| 3 | simp2r 1200 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β πΊ β π) | |
| 4 | trlcoat.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 5 | trlcoat.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
| 6 | 4, 5 | ltrncnv 38806 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΊ β π) β β‘πΊ β π) |
| 7 | 1, 3, 6 | syl2anc 584 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β β‘πΊ β π) |
| 8 | simp3 1138 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β (π βπΉ) β (π βπΊ)) | |
| 9 | trlcoat.r | . . . . 5 β’ π = ((trLβπΎ)βπ) | |
| 10 | 4, 5, 9 | trlcnv 38825 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΊ β π) β (π ββ‘πΊ) = (π βπΊ)) |
| 11 | 1, 3, 10 | syl2anc 584 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β (π ββ‘πΊ) = (π βπΊ)) |
| 12 | 8, 11 | neeqtrrd 3014 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β (π βπΉ) β (π ββ‘πΊ)) |
| 13 | trlcoat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
| 14 | 13, 4, 5, 9 | trlcoat 39383 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ β‘πΊ β π) β§ (π βπΉ) β (π ββ‘πΊ)) β (π β(πΉ β β‘πΊ)) β π΄) |
| 15 | 1, 2, 7, 12, 14 | syl121anc 1375 | 1 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π βπΉ) β (π βπΊ)) β (π β(πΉ β β‘πΊ)) β π΄) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2939 β‘ccnv 5665 β ccom 5670 βcfv 6529 Atomscatm 37922 HLchlt 38009 LHypclh 38644 LTrncltrn 38761 trLctrl 38818 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-riotaBAD 37612 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 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-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7954 df-2nd 7955 df-undef 8237 df-map 8802 df-proset 18227 df-poset 18245 df-plt 18262 df-lub 18278 df-glb 18279 df-join 18280 df-meet 18281 df-p0 18357 df-p1 18358 df-lat 18364 df-clat 18431 df-oposet 37835 df-ol 37837 df-oml 37838 df-covers 37925 df-ats 37926 df-atl 37957 df-cvlat 37981 df-hlat 38010 df-llines 38158 df-lplanes 38159 df-lvols 38160 df-lines 38161 df-psubsp 38163 df-pmap 38164 df-padd 38456 df-lhyp 38648 df-laut 38649 df-ldil 38764 df-ltrn 38765 df-trl 38819 |
| This theorem is referenced by: cdlemh1 39475 cdlemk3 39493 cdlemk6 39497 cdlemk7 39508 cdlemk12 39510 cdlemkole 39513 cdlemk14 39514 cdlemk15 39515 cdlemk5u 39521 cdlemk6u 39522 cdlemk7u 39530 cdlemk12u 39532 cdlemkfid1N 39581 |
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