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Mirrors > Home > MPE Home > Th. List > mircom | Structured version Visualization version GIF version |
Description: Variation on mirmir 28180. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
Ref | Expression |
---|---|
mirval.p | β’ π = (BaseβπΊ) |
mirval.d | β’ β = (distβπΊ) |
mirval.i | β’ πΌ = (ItvβπΊ) |
mirval.l | β’ πΏ = (LineGβπΊ) |
mirval.s | β’ π = (pInvGβπΊ) |
mirval.g | β’ (π β πΊ β TarskiG) |
mirval.a | β’ (π β π΄ β π) |
mirfv.m | β’ π = (πβπ΄) |
mirmir.b | β’ (π β π΅ β π) |
mircom.1 | β’ (π β (πβπ΅) = πΆ) |
Ref | Expression |
---|---|
mircom | β’ (π β (πβπΆ) = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mircom.1 | . . 3 β’ (π β (πβπ΅) = πΆ) | |
2 | 1 | fveq2d 6894 | . 2 β’ (π β (πβ(πβπ΅)) = (πβπΆ)) |
3 | mirval.p | . . 3 β’ π = (BaseβπΊ) | |
4 | mirval.d | . . 3 β’ β = (distβπΊ) | |
5 | mirval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
6 | mirval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
7 | mirval.s | . . 3 β’ π = (pInvGβπΊ) | |
8 | mirval.g | . . 3 β’ (π β πΊ β TarskiG) | |
9 | mirval.a | . . 3 β’ (π β π΄ β π) | |
10 | mirfv.m | . . 3 β’ π = (πβπ΄) | |
11 | mirmir.b | . . 3 β’ (π β π΅ β π) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | mirmir 28180 | . 2 β’ (π β (πβ(πβπ΅)) = π΅) |
13 | 2, 12 | eqtr3d 2772 | 1 β’ (π β (πβπΆ) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βcfv 6542 Basecbs 17148 distcds 17210 TarskiGcstrkg 27945 Itvcitv 27951 LineGclng 27952 pInvGcmir 28170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-trkgc 27966 df-trkgb 27967 df-trkgcb 27968 df-trkg 27971 df-mir 28171 |
This theorem is referenced by: miduniq 28203 colperpexlem3 28250 mideulem2 28252 midex 28255 opphllem1 28265 opphllem2 28266 opphllem3 28267 opphllem5 28269 opphllem6 28270 trgcopyeulem 28323 |
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