![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2737 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | dia1.h | . . . . 5 β’ π» = (LHypβπΎ) | |
4 | 2, 3 | lhpbase 38464 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
5 | 4 | adantl 483 | . . 3 β’ ((πΎ β HL β§ π β π») β π β (BaseβπΎ)) |
6 | hllat 37828 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
7 | eqid 2737 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
8 | 2, 7 | latref 18331 | . . . 4 β’ ((πΎ β Lat β§ π β (BaseβπΎ)) β π(leβπΎ)π) |
9 | 6, 4, 8 | syl2an 597 | . . 3 β’ ((πΎ β HL β§ π β π») β π(leβπΎ)π) |
10 | dia1.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
11 | eqid 2737 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
12 | dia1.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
13 | 2, 7, 3, 10, 11, 12 | diaval 39498 | . . 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 38650 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π) β (((trLβπΎ)βπ)βπ)(leβπΎ)π) |
16 | 15 | ralrimiva 3144 | . . 3 β’ ((πΎ β HL β§ π β π») β βπ β π (((trLβπΎ)βπ)βπ)(leβπΎ)π) |
17 | rabid2 3437 | . . 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 2780 | 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 397 = wceq 1542 β wcel 2107 βwral 3065 {crab 3408 class class class wbr 5106 βcfv 6497 Basecbs 17084 lecple 17141 Latclat 18321 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 DIsoAcdia 39494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-disoa 39495 |
This theorem is referenced by: dia1elN 39520 |
Copyright terms: Public domain | W3C validator |