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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 39626 | . . 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 2815 | . 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 6896 | . . . . . 6 β’ (π = πΉ β (πβπ₯) = (πΉβπ₯)) | |
9 | 8 | eqeq1d 2730 | . . . . 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 3174 | . . 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 3682 | . 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 1534 β wcel 2099 βwral 3058 {crab 3429 β wss 3947 βcfv 6548 Atomscatm 38735 PSubSpcpsubsp 38969 WAtomscwpointsN 39459 PAutcpautN 39460 DilcdilN 39575 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-dilN 39579 |
This theorem is referenced by: (None) |
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