![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihordlem7 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.) |
Ref | Expression |
---|---|
dihordlem8.b | β’ π΅ = (BaseβπΎ) |
dihordlem8.l | β’ β€ = (leβπΎ) |
dihordlem8.a | β’ π΄ = (AtomsβπΎ) |
dihordlem8.h | β’ π» = (LHypβπΎ) |
dihordlem8.p | β’ π = ((ocβπΎ)βπ) |
dihordlem8.o | β’ π = (β β π β¦ ( I βΎ π΅)) |
dihordlem8.t | β’ π = ((LTrnβπΎ)βπ) |
dihordlem8.e | β’ πΈ = ((TEndoβπΎ)βπ) |
dihordlem8.u | β’ π = ((DVecHβπΎ)βπ) |
dihordlem8.s | β’ + = (+gβπ) |
dihordlem8.g | β’ πΊ = (β©β β π (ββπ) = π ) |
Ref | Expression |
---|---|
dihordlem7 | β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β (π = ((π βπΊ) β π) β§ π = π )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp33 1208 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©)) | |
2 | simp1 1133 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β (πΎ β HL β§ π β π»)) | |
3 | simp2l 1196 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β (π β π΄ β§ Β¬ π β€ π)) | |
4 | simp2r 1197 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β (π β π΄ β§ Β¬ π β€ π)) | |
5 | simp31 1206 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β π β πΈ) | |
6 | simp32 1207 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β π β π) | |
7 | dihordlem8.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
8 | dihordlem8.l | . . . . 5 β’ β€ = (leβπΎ) | |
9 | dihordlem8.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
10 | dihordlem8.h | . . . . 5 β’ π» = (LHypβπΎ) | |
11 | dihordlem8.p | . . . . 5 β’ π = ((ocβπΎ)βπ) | |
12 | dihordlem8.o | . . . . 5 β’ π = (β β π β¦ ( I βΎ π΅)) | |
13 | dihordlem8.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
14 | dihordlem8.e | . . . . 5 β’ πΈ = ((TEndoβπΎ)βπ) | |
15 | dihordlem8.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
16 | dihordlem8.s | . . . . 5 β’ + = (+gβπ) | |
17 | dihordlem8.g | . . . . 5 β’ πΊ = (β©β β π (ββπ) = π ) | |
18 | 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | dihordlem6 40597 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π)) β (β¨(π βπΊ), π β© + β¨π, πβ©) = β¨((π βπΊ) β π), π β©) |
19 | 2, 3, 4, 5, 6, 18 | syl122anc 1376 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β (β¨(π βπΊ), π β© + β¨π, πβ©) = β¨((π βπΊ) β π), π β©) |
20 | 1, 19 | eqtrd 2766 | . 2 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β β¨π, πβ© = β¨((π βπΊ) β π), π β©) |
21 | fvex 6898 | . . . 4 β’ (π βπΊ) β V | |
22 | vex 3472 | . . . 4 β’ π β V | |
23 | 21, 22 | coex 7920 | . . 3 β’ ((π βπΊ) β π) β V |
24 | vex 3472 | . . 3 β’ π β V | |
25 | 23, 24 | opth2 5473 | . 2 β’ (β¨π, πβ© = β¨((π βπΊ) β π), π β© β (π = ((π βπΊ) β π) β§ π = π )) |
26 | 20, 25 | sylib 217 | 1 β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β πΈ β§ π β π β§ β¨π, πβ© = (β¨(π βπΊ), π β© + β¨π, πβ©))) β (π = ((π βπΊ) β π) β§ π = π )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β¨cop 4629 class class class wbr 5141 β¦ cmpt 5224 I cid 5566 βΎ cres 5671 β ccom 5673 βcfv 6537 β©crio 7360 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 lecple 17213 occoc 17214 Atomscatm 38646 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 TEndoctendo 40136 DVecHcdvh 40462 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 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-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tendo 40139 df-edring 40141 df-dvech 40463 |
This theorem is referenced by: dihordlem7b 40599 |
Copyright terms: Public domain | W3C validator |