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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > diael | Structured version Visualization version GIF version |
Description: A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014.) |
Ref | Expression |
---|---|
diass.b | β’ π΅ = (BaseβπΎ) |
diass.l | β’ β€ = (leβπΎ) |
diass.h | β’ π» = (LHypβπΎ) |
diass.t | β’ π = ((LTrnβπΎ)βπ) |
diass.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
diael | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ πΉ β (πΌβπ)) β πΉ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diass.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | diass.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | diass.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | diass.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
5 | diass.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | diass 40403 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β π) |
7 | 6 | sseld 3973 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΉ β (πΌβπ) β πΉ β π)) |
8 | 7 | 3impia 1114 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ πΉ β (πΌβπ)) β πΉ 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 5138 βcfv 6533 Basecbs 17143 lecple 17203 LHypclh 39345 LTrncltrn 39462 DIsoAcdia 40389 |
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 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-disoa 40390 |
This theorem is referenced by: dialss 40407 dibelval1st1 40511 diblsmopel 40532 |
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