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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isdilN | Structured version Visualization version GIF version |
Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dilset.a | β’ π΄ = (AtomsβπΎ) |
dilset.s | β’ π = (PSubSpβπΎ) |
dilset.w | β’ π = (WAtomsβπΎ) |
dilset.m | β’ π = (PAutβπΎ) |
dilset.l | β’ πΏ = (DilβπΎ) |
Ref | Expression |
---|---|
isdilN | β’ ((πΎ β π΅ β§ π· β π΄) β (πΉ β (πΏβπ·) β (πΉ β π β§ βπ₯ β π (π₯ β (πβπ·) β (πΉβπ₯) = π₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dilset.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | dilset.s | . . . 4 β’ π = (PSubSpβπΎ) | |
3 | dilset.w | . . . 4 β’ π = (WAtomsβπΎ) | |
4 | dilset.m | . . . 4 β’ π = (PAutβπΎ) | |
5 | dilset.l | . . . 4 β’ πΏ = (DilβπΎ) | |
6 | 1, 2, 3, 4, 5 | dilsetN 39535 | . . 3 β’ ((πΎ β π΅ β§ π· β π΄) β (πΏβπ·) = {π β π β£ βπ₯ β π (π₯ β (πβπ·) β (πβπ₯) = π₯)}) |
7 | 6 | eleq2d 2813 | . 2 β’ ((πΎ β π΅ β§ π· β π΄) β (πΉ β (πΏβπ·) β πΉ β {π β π β£ βπ₯ β π (π₯ β (πβπ·) β (πβπ₯) = π₯)})) |
8 | fveq1 6883 | . . . . . 6 β’ (π = πΉ β (πβπ₯) = (πΉβπ₯)) | |
9 | 8 | eqeq1d 2728 | . . . . 5 β’ (π = πΉ β ((πβπ₯) = π₯ β (πΉβπ₯) = π₯)) |
10 | 9 | imbi2d 340 | . . . 4 β’ (π = πΉ β ((π₯ β (πβπ·) β (πβπ₯) = π₯) β (π₯ β (πβπ·) β (πΉβπ₯) = π₯))) |
11 | 10 | ralbidv 3171 | . . 3 β’ (π = πΉ β (βπ₯ β π (π₯ β (πβπ·) β (πβπ₯) = π₯) β βπ₯ β π (π₯ β (πβπ·) β (πΉβπ₯) = π₯))) |
12 | 11 | elrab 3678 | . 2 β’ (πΉ β {π β π β£ βπ₯ β π (π₯ β (πβπ·) β (πβπ₯) = π₯)} β (πΉ β π β§ βπ₯ β π (π₯ β (πβπ·) β (πΉβπ₯) = π₯))) |
13 | 7, 12 | bitrdi 287 | 1 β’ ((πΎ β π΅ β§ π· β π΄) β (πΉ β (πΏβπ·) β (πΉ β π β§ βπ₯ β π (π₯ β (πβπ·) β (πΉβπ₯) = π₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 β wss 3943 βcfv 6536 Atomscatm 38644 PSubSpcpsubsp 38878 WAtomscwpointsN 39368 PAutcpautN 39369 DilcdilN 39484 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-dilN 39488 |
This theorem is referenced by: (None) |
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