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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dian0 | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.) |
Ref | Expression |
---|---|
dian0.b | β’ π΅ = (BaseβπΎ) |
dian0.l | β’ β€ = (leβπΎ) |
dian0.h | β’ π» = (LHypβπΎ) |
dian0.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dian0 | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dian0.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | dian0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
3 | eqid 2724 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
4 | 1, 2, 3 | idltrn 39524 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
5 | 4 | adantr 480 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
6 | eqid 2724 | . . . . . 6 β’ (0.βπΎ) = (0.βπΎ) | |
7 | eqid 2724 | . . . . . 6 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
8 | 1, 6, 2, 7 | trlid0 39550 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (((trLβπΎ)βπ)β( I βΎ π΅)) = (0.βπΎ)) |
9 | 8 | adantr 480 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (((trLβπΎ)βπ)β( I βΎ π΅)) = (0.βπΎ)) |
10 | hlatl 38733 | . . . . . 6 β’ (πΎ β HL β πΎ β AtLat) | |
11 | 10 | adantr 480 | . . . . 5 β’ ((πΎ β HL β§ π β π») β πΎ β AtLat) |
12 | simpl 482 | . . . . 5 β’ ((π β π΅ β§ π β€ π) β π β π΅) | |
13 | dian0.l | . . . . . 6 β’ β€ = (leβπΎ) | |
14 | 1, 13, 6 | atl0le 38677 | . . . . 5 β’ ((πΎ β AtLat β§ π β π΅) β (0.βπΎ) β€ π) |
15 | 11, 12, 14 | syl2an 595 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (0.βπΎ) β€ π) |
16 | 9, 15 | eqbrtrd 5161 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (((trLβπΎ)βπ)β( I βΎ π΅)) β€ π) |
17 | dian0.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
18 | 1, 13, 2, 3, 7, 17 | diaelval 40407 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (( I βΎ π΅) β (πΌβπ) β (( I βΎ π΅) β ((LTrnβπΎ)βπ) β§ (((trLβπΎ)βπ)β( I βΎ π΅)) β€ π))) |
19 | 5, 16, 18 | mpbir2and 710 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β ( I βΎ π΅) β (πΌβπ)) |
20 | 19 | ne0d 4328 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 β c0 4315 class class class wbr 5139 I cid 5564 βΎ cres 5669 βcfv 6534 Basecbs 17149 lecple 17209 0.cp0 18384 AtLatcal 38637 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-clat 18460 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: dialss 40420 dibn0 40527 |
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