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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > diatrl | Structured version Visualization version GIF version |
Description: Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.) |
Ref | Expression |
---|---|
diatrl.b | β’ π΅ = (BaseβπΎ) |
diatrl.l | β’ β€ = (leβπΎ) |
diatrl.h | β’ π» = (LHypβπΎ) |
diatrl.t | β’ π = ((LTrnβπΎ)βπ) |
diatrl.r | β’ π = ((trLβπΎ)βπ) |
diatrl.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
diatrl | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ πΉ β (πΌβπ)) β (π βπΉ) β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diatrl.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | diatrl.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | diatrl.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | diatrl.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
5 | diatrl.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
6 | diatrl.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
7 | 1, 2, 3, 4, 5, 6 | diaelval 40398 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΉ β (πΌβπ) β (πΉ β π β§ (π βπΉ) β€ π))) |
8 | simpr 484 | . . 3 β’ ((πΉ β π β§ (π βπΉ) β€ π) β (π βπΉ) β€ π) | |
9 | 7, 8 | syl6bi 253 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΉ β (πΌβπ) β (π βπΉ) β€ π)) |
10 | 9 | 3impia 1114 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ πΉ β (πΌβπ)) β (π βπΉ) β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 Basecbs 17145 lecple 17205 LHypclh 39349 LTrncltrn 39466 trLctrl 39523 DIsoAcdia 40393 |
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-pr 5418 |
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-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-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-disoa 40394 |
This theorem is referenced by: dialss 40411 dibelval1st2N 40516 diblss 40535 |
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