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Mirrors > Home > MPE Home > Th. List > colperpexlem2 | Structured version Visualization version GIF version |
Description: Lemma for colperpex 28576. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
Ref | Expression |
---|---|
colperpex.p | β’ π = (BaseβπΊ) |
colperpex.d | β’ β = (distβπΊ) |
colperpex.i | β’ πΌ = (ItvβπΊ) |
colperpex.l | β’ πΏ = (LineGβπΊ) |
colperpex.g | β’ (π β πΊ β TarskiG) |
colperpexlem.s | β’ π = (pInvGβπΊ) |
colperpexlem.m | β’ π = (πβπ΄) |
colperpexlem.n | β’ π = (πβπ΅) |
colperpexlem.k | β’ πΎ = (πβπ) |
colperpexlem.a | β’ (π β π΄ β π) |
colperpexlem.b | β’ (π β π΅ β π) |
colperpexlem.c | β’ (π β πΆ β π) |
colperpexlem.q | β’ (π β π β π) |
colperpexlem.1 | β’ (π β β¨βπ΄π΅πΆββ© β (βGβπΊ)) |
colperpexlem.2 | β’ (π β (πΎβ(πβπΆ)) = (πβπΆ)) |
colperpexlem2.e | β’ (π β π΅ β πΆ) |
Ref | Expression |
---|---|
colperpexlem2 | β’ (π β π΄ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | colperpexlem2.e | . . 3 β’ (π β π΅ β πΆ) | |
2 | simpr 483 | . . . . . . . . . 10 β’ ((π β§ π΄ = π) β π΄ = π) | |
3 | 2 | fveq2d 6894 | . . . . . . . . 9 β’ ((π β§ π΄ = π) β (πβπ΄) = (πβπ)) |
4 | colperpexlem.m | . . . . . . . . 9 β’ π = (πβπ΄) | |
5 | colperpexlem.k | . . . . . . . . 9 β’ πΎ = (πβπ) | |
6 | 3, 4, 5 | 3eqtr4g 2790 | . . . . . . . 8 β’ ((π β§ π΄ = π) β π = πΎ) |
7 | 6 | fveq1d 6892 | . . . . . . 7 β’ ((π β§ π΄ = π) β (πβ(πβπΆ)) = (πΎβ(πβπΆ))) |
8 | colperpex.p | . . . . . . . . 9 β’ π = (BaseβπΊ) | |
9 | colperpex.d | . . . . . . . . 9 β’ β = (distβπΊ) | |
10 | colperpex.i | . . . . . . . . 9 β’ πΌ = (ItvβπΊ) | |
11 | colperpex.l | . . . . . . . . 9 β’ πΏ = (LineGβπΊ) | |
12 | colperpexlem.s | . . . . . . . . 9 β’ π = (pInvGβπΊ) | |
13 | colperpex.g | . . . . . . . . 9 β’ (π β πΊ β TarskiG) | |
14 | colperpexlem.a | . . . . . . . . 9 β’ (π β π΄ β π) | |
15 | colperpexlem.c | . . . . . . . . 9 β’ (π β πΆ β π) | |
16 | 8, 9, 10, 11, 12, 13, 14, 4, 15 | mirmir 28505 | . . . . . . . 8 β’ (π β (πβ(πβπΆ)) = πΆ) |
17 | 16 | adantr 479 | . . . . . . 7 β’ ((π β§ π΄ = π) β (πβ(πβπΆ)) = πΆ) |
18 | colperpexlem.2 | . . . . . . . 8 β’ (π β (πΎβ(πβπΆ)) = (πβπΆ)) | |
19 | 18 | adantr 479 | . . . . . . 7 β’ ((π β§ π΄ = π) β (πΎβ(πβπΆ)) = (πβπΆ)) |
20 | 7, 17, 19 | 3eqtr3rd 2774 | . . . . . 6 β’ ((π β§ π΄ = π) β (πβπΆ) = πΆ) |
21 | colperpexlem.b | . . . . . . . 8 β’ (π β π΅ β π) | |
22 | colperpexlem.n | . . . . . . . 8 β’ π = (πβπ΅) | |
23 | 8, 9, 10, 11, 12, 13, 21, 22, 15 | mirinv 28509 | . . . . . . 7 β’ (π β ((πβπΆ) = πΆ β π΅ = πΆ)) |
24 | 23 | adantr 479 | . . . . . 6 β’ ((π β§ π΄ = π) β ((πβπΆ) = πΆ β π΅ = πΆ)) |
25 | 20, 24 | mpbid 231 | . . . . 5 β’ ((π β§ π΄ = π) β π΅ = πΆ) |
26 | 25 | ex 411 | . . . 4 β’ (π β (π΄ = π β π΅ = πΆ)) |
27 | 26 | necon3ad 2943 | . . 3 β’ (π β (π΅ β πΆ β Β¬ π΄ = π)) |
28 | 1, 27 | mpd 15 | . 2 β’ (π β Β¬ π΄ = π) |
29 | 28 | neqned 2937 | 1 β’ (π β π΄ β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 βcfv 6543 β¨βcs3 14820 Basecbs 17174 distcds 17236 TarskiGcstrkg 28270 Itvcitv 28276 LineGclng 28277 pInvGcmir 28495 βGcrag 28536 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 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-riota 7369 df-ov 7416 df-trkgc 28291 df-trkgb 28292 df-trkgcb 28293 df-trkg 28296 df-mir 28496 |
This theorem is referenced by: colperpexlem3 28575 |
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