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Mirrors > Home > MPE Home > Th. List > colperpexlem2 | Structured version Visualization version GIF version |
Description: Lemma for colperpex 28524. 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 484 | . . . . . . . . . 10 β’ ((π β§ π΄ = π) β π΄ = π) | |
3 | 2 | fveq2d 6895 | . . . . . . . . 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 2792 | . . . . . . . 8 β’ ((π β§ π΄ = π) β π = πΎ) |
7 | 6 | fveq1d 6893 | . . . . . . 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 28453 | . . . . . . . 8 β’ (π β (πβ(πβπΆ)) = πΆ) |
17 | 16 | adantr 480 | . . . . . . 7 β’ ((π β§ π΄ = π) β (πβ(πβπΆ)) = πΆ) |
18 | colperpexlem.2 | . . . . . . . 8 β’ (π β (πΎβ(πβπΆ)) = (πβπΆ)) | |
19 | 18 | adantr 480 | . . . . . . 7 β’ ((π β§ π΄ = π) β (πΎβ(πβπΆ)) = (πβπΆ)) |
20 | 7, 17, 19 | 3eqtr3rd 2776 | . . . . . 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 28457 | . . . . . . 7 β’ (π β ((πβπΆ) = πΆ β π΅ = πΆ)) |
24 | 23 | adantr 480 | . . . . . 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 412 | . . . 4 β’ (π β (π΄ = π β π΅ = πΆ)) |
27 | 26 | necon3ad 2948 | . . 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 2942 | 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 395 = wceq 1534 β wcel 2099 β wne 2935 βcfv 6542 β¨βcs3 14817 Basecbs 17171 distcds 17233 TarskiGcstrkg 28218 Itvcitv 28224 LineGclng 28225 pInvGcmir 28443 βGcrag 28484 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-trkgc 28239 df-trkgb 28240 df-trkgcb 28241 df-trkg 28244 df-mir 28444 |
This theorem is referenced by: colperpexlem3 28523 |
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