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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 40084. (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 1133 | . 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 1196 | . 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 1197 | . . 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 39507 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΊ β π) β β‘πΊ β π) |
| 7 | 1, 3, 6 | syl2anc 583 | . 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 1135 | . . 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 39526 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΊ β π) β (π ββ‘πΊ) = (π βπΊ)) |
| 11 | 1, 3, 10 | syl2anc 583 | . . 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 3007 | . 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 40084 | . 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 1372 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 β‘ccnv 5665 β ccom 5670 βcfv 6533 Atomscatm 38623 HLchlt 38710 LHypclh 39345 LTrncltrn 39462 trLctrl 39519 |
| 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-pow 5353 ax-pr 5417 ax-un 7718 ax-riotaBAD 38313 |
| 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-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 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 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-1st 7968 df-2nd 7969 df-undef 8253 df-map 8818 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-llines 38859 df-lplanes 38860 df-lvols 38861 df-lines 38862 df-psubsp 38864 df-pmap 38865 df-padd 39157 df-lhyp 39349 df-laut 39350 df-ldil 39465 df-ltrn 39466 df-trl 39520 |
| This theorem is referenced by: cdlemh1 40176 cdlemk3 40194 cdlemk6 40198 cdlemk7 40209 cdlemk12 40211 cdlemkole 40214 cdlemk14 40215 cdlemk15 40216 cdlemk5u 40222 cdlemk6u 40223 cdlemk7u 40231 cdlemk12u 40233 cdlemkfid1N 40282 |
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