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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 2724 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | dia1.h | . . . . 5 β’ π» = (LHypβπΎ) | |
4 | 2, 3 | lhpbase 39372 | . . . 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 38736 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
7 | eqid 2724 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
8 | 2, 7 | latref 18402 | . . . 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 2724 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
12 | dia1.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
13 | 2, 7, 3, 10, 11, 12 | diaval 40406 | . . 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 834 | . 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 39558 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π) β (((trLβπΎ)βπ)βπ)(leβπΎ)π) |
16 | 15 | ralrimiva 3138 | . . 3 β’ ((πΎ β HL β§ π β π») β βπ β π (((trLβπΎ)βπ)βπ)(leβπΎ)π) |
17 | rabid2 3456 | . . 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 2767 | 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 1533 β wcel 2098 βwral 3053 {crab 3424 class class class wbr 5139 βcfv 6534 Basecbs 17149 lecple 17209 Latclat 18392 HLchlt 38723 LHypclh 39358 LTrncltrn 39475 trLctrl 39532 DIsoAcdia 40402 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-atl 38671 df-cvlat 38695 df-hlat 38724 df-lhyp 39362 df-laut 39363 df-ldil 39478 df-ltrn 39479 df-trl 39533 df-disoa 40403 |
This theorem is referenced by: dia1elN 40428 |
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