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Mirrors > Home > MPE Home > Th. List > mirne | Structured version Visualization version GIF version |
Description: Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020.) |
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΄) |
mirinv.b | β’ (π β π΅ β π) |
mirne.1 | β’ (π β π΅ β π΄) |
Ref | Expression |
---|---|
mirne | β’ (π β (πβπ΅) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . . . 5 β’ ((π β§ (πβπ΅) = π΄) β (πβπ΅) = π΄) | |
2 | 1 | fveq2d 6896 | . . . 4 β’ ((π β§ (πβπ΅) = π΄) β (πβ(πβπ΅)) = (πβπ΄)) |
3 | mirval.p | . . . . . 6 β’ π = (BaseβπΊ) | |
4 | mirval.d | . . . . . 6 β’ β = (distβπΊ) | |
5 | mirval.i | . . . . . 6 β’ πΌ = (ItvβπΊ) | |
6 | mirval.l | . . . . . 6 β’ πΏ = (LineGβπΊ) | |
7 | mirval.s | . . . . . 6 β’ π = (pInvGβπΊ) | |
8 | mirval.g | . . . . . 6 β’ (π β πΊ β TarskiG) | |
9 | mirval.a | . . . . . 6 β’ (π β π΄ β π) | |
10 | mirfv.m | . . . . . 6 β’ π = (πβπ΄) | |
11 | mirinv.b | . . . . . 6 β’ (π β π΅ β π) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | mirmir 28178 | . . . . 5 β’ (π β (πβ(πβπ΅)) = π΅) |
13 | 12 | adantr 479 | . . . 4 β’ ((π β§ (πβπ΅) = π΄) β (πβ(πβπ΅)) = π΅) |
14 | eqid 2730 | . . . . . 6 β’ π΄ = π΄ | |
15 | 3, 4, 5, 6, 7, 8, 9, 10, 9 | mirinv 28182 | . . . . . 6 β’ (π β ((πβπ΄) = π΄ β π΄ = π΄)) |
16 | 14, 15 | mpbiri 257 | . . . . 5 β’ (π β (πβπ΄) = π΄) |
17 | 16 | adantr 479 | . . . 4 β’ ((π β§ (πβπ΅) = π΄) β (πβπ΄) = π΄) |
18 | 2, 13, 17 | 3eqtr3d 2778 | . . 3 β’ ((π β§ (πβπ΅) = π΄) β π΅ = π΄) |
19 | mirne.1 | . . . . 5 β’ (π β π΅ β π΄) | |
20 | 19 | adantr 479 | . . . 4 β’ ((π β§ (πβπ΅) = π΄) β π΅ β π΄) |
21 | 20 | neneqd 2943 | . . 3 β’ ((π β§ (πβπ΅) = π΄) β Β¬ π΅ = π΄) |
22 | 18, 21 | pm2.65da 813 | . 2 β’ (π β Β¬ (πβπ΅) = π΄) |
23 | 22 | neqned 2945 | 1 β’ (π β (πβπ΅) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wne 2938 βcfv 6544 Basecbs 17150 distcds 17212 TarskiGcstrkg 27943 Itvcitv 27949 LineGclng 27950 pInvGcmir 28168 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-trkgc 27964 df-trkgb 27965 df-trkgcb 27966 df-trkg 27969 df-mir 28169 |
This theorem is referenced by: mirhl2 28197 sacgr 28347 |
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