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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia1N | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia1.h | β’ π» = (LHypβπΎ) |
dia1.t | β’ π = ((LTrnβπΎ)βπ) |
dia1.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dia1N | β’ ((πΎ β HL β§ π β π») β (πΌβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 β’ ((πΎ β HL β§ π β π») β (πΎ β HL β§ π β π»)) | |
2 | eqid 2728 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | dia1.h | . . . . 5 β’ π» = (LHypβπΎ) | |
4 | 2, 3 | lhpbase 39471 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
5 | 4 | adantl 481 | . . 3 β’ ((πΎ β HL β§ π β π») β π β (BaseβπΎ)) |
6 | hllat 38835 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
7 | eqid 2728 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
8 | 2, 7 | latref 18432 | . . . 4 β’ ((πΎ β Lat β§ π β (BaseβπΎ)) β π(leβπΎ)π) |
9 | 6, 4, 8 | syl2an 595 | . . 3 β’ ((πΎ β HL β§ π β π») β π(leβπΎ)π) |
10 | dia1.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
11 | eqid 2728 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
12 | dia1.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
13 | 2, 7, 3, 10, 11, 12 | diaval 40505 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π(leβπΎ)π)) β (πΌβπ) = {π β π β£ (((trLβπΎ)βπ)βπ)(leβπΎ)π}) |
14 | 1, 5, 9, 13 | syl12anc 836 | . 2 β’ ((πΎ β HL β§ π β π») β (πΌβπ) = {π β π β£ (((trLβπΎ)βπ)βπ)(leβπΎ)π}) |
15 | 7, 3, 10, 11 | trlle 39657 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π) β (((trLβπΎ)βπ)βπ)(leβπΎ)π) |
16 | 15 | ralrimiva 3143 | . . 3 β’ ((πΎ β HL β§ π β π») β βπ β π (((trLβπΎ)βπ)βπ)(leβπΎ)π) |
17 | rabid2 3461 | . . 3 β’ (π = {π β π β£ (((trLβπΎ)βπ)βπ)(leβπΎ)π} β βπ β π (((trLβπΎ)βπ)βπ)(leβπΎ)π) | |
18 | 16, 17 | sylibr 233 | . 2 β’ ((πΎ β HL β§ π β π») β π = {π β π β£ (((trLβπΎ)βπ)βπ)(leβπΎ)π}) |
19 | 14, 18 | eqtr4d 2771 | 1 β’ ((πΎ β HL β§ π β π») β (πΌβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3058 {crab 3429 class class class wbr 5148 βcfv 6548 Basecbs 17179 lecple 17239 Latclat 18422 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 trLctrl 39631 DIsoAcdia 40501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-disoa 40502 |
This theorem is referenced by: dia1elN 40527 |
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