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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 40501 | . . 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 1115 | 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 1085 = wceq 1534 β wcel 2099 class class class wbr 5143 βcfv 6543 Basecbs 17174 lecple 17234 LHypclh 39452 LTrncltrn 39569 trLctrl 39626 DIsoAcdia 40496 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
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 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-disoa 40497 |
This theorem is referenced by: dialss 40514 dibelval1st2N 40619 diblss 40638 |
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