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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemn7 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.) |
Ref | Expression |
---|---|
cdlemn8.b | β’ π΅ = (BaseβπΎ) |
cdlemn8.l | β’ β€ = (leβπΎ) |
cdlemn8.a | β’ π΄ = (AtomsβπΎ) |
cdlemn8.h | β’ π» = (LHypβπΎ) |
cdlemn8.p | β’ π = ((ocβπΎ)βπ) |
cdlemn8.o | β’ π = (β β π β¦ ( I βΎ π΅)) |
cdlemn8.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemn8.e | β’ πΈ = ((TEndoβπΎ)βπ) |
cdlemn8.u | β’ π = ((DVecHβπΎ)βπ) |
cdlemn8.s | β’ + = (+gβπ) |
cdlemn8.f | β’ πΉ = (β©β β π (ββπ) = π) |
cdlemn8.g | β’ πΊ = (β©β β π (ββπ) = π ) |
Ref | Expression |
---|---|
cdlemn7 | 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β (πΊ = ((π βπΉ) β π) β§ ( I βΎ π) = π )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp33 1211 | . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β β¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©)) | |
2 | simp1 1136 | . . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β (πΎ β HL β§ π β π»)) | |
3 | simp2l 1199 | . . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β (π β π΄ β§ Β¬ π β€ π)) | |
4 | simp2r 1200 | . . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β (π β π΄ β§ Β¬ π β€ π)) | |
5 | simp31 1209 | . . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β π β πΈ) | |
6 | simp32 1210 | . . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β π β π) | |
7 | cdlemn8.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
8 | cdlemn8.l | . . . . 5 β’ β€ = (leβπΎ) | |
9 | cdlemn8.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
10 | cdlemn8.h | . . . . 5 β’ π» = (LHypβπΎ) | |
11 | cdlemn8.p | . . . . 5 β’ π = ((ocβπΎ)βπ) | |
12 | cdlemn8.o | . . . . 5 β’ π = (β β π β¦ ( I βΎ π΅)) | |
13 | cdlemn8.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
14 | cdlemn8.e | . . . . 5 β’ πΈ = ((TEndoβπΎ)βπ) | |
15 | cdlemn8.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
16 | cdlemn8.s | . . . . 5 β’ + = (+gβπ) | |
17 | cdlemn8.f | . . . . 5 β’ πΉ = (β©β β π (ββπ) = π) | |
18 | 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | cdlemn6 40376 | . . . 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 βπΉ), π β© + β¨π, πβ©) = β¨((π βπΉ) β π), π β©) |
19 | 2, 3, 4, 5, 6, 18 | syl122anc 1379 | . . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β (β¨(π βπΉ), π β© + β¨π, πβ©) = β¨((π βπΉ) β π), π β©) |
20 | 1, 19 | eqtrd 2772 | . 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β β¨πΊ, ( I βΎ π)β© = β¨((π βπΉ) β π), π β©) |
21 | fvex 6904 | . . . 4 β’ (π βπΉ) β V | |
22 | vex 3478 | . . . 4 β’ π β V | |
23 | 21, 22 | coex 7923 | . . 3 β’ ((π βπΉ) β π) β V |
24 | vex 3478 | . . 3 β’ π β V | |
25 | 23, 24 | opth2 5480 | . 2 β’ (β¨πΊ, ( I βΎ π)β© = β¨((π βπΉ) β π), π β© β (πΊ = ((π βπΉ) β π) β§ ( I βΎ 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¨πΊ, ( I βΎ π)β© = (β¨(π βπΉ), π β© + β¨π, πβ©))) β (πΊ = ((π βπΉ) β π) β§ ( I βΎ π) = π )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β¨cop 4634 class class class wbr 5148 β¦ cmpt 5231 I cid 5573 βΎ cres 5678 β ccom 5680 βcfv 6543 β©crio 7366 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 lecple 17208 occoc 17209 Atomscatm 38436 HLchlt 38523 LHypclh 39158 LTrncltrn 39275 TEndoctendo 39926 DVecHcdvh 40252 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 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 38126 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 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 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tendo 39929 df-edring 39931 df-dvech 40253 |
This theorem is referenced by: cdlemn8 40378 |
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