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Mirrors > Home > MPE Home > Th. List > tgcgrneq | Structured version Visualization version GIF version |
Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
Ref | Expression |
---|---|
tkgeom.p | β’ π = (BaseβπΊ) |
tkgeom.d | β’ β = (distβπΊ) |
tkgeom.i | β’ πΌ = (ItvβπΊ) |
tkgeom.g | β’ (π β πΊ β TarskiG) |
tgcgrcomlr.a | β’ (π β π΄ β π) |
tgcgrcomlr.b | β’ (π β π΅ β π) |
tgcgrcomlr.c | β’ (π β πΆ β π) |
tgcgrcomlr.d | β’ (π β π· β π) |
tgcgrcomlr.6 | β’ (π β (π΄ β π΅) = (πΆ β π·)) |
tgcgrneq.1 | β’ (π β π΄ β π΅) |
Ref | Expression |
---|---|
tgcgrneq | β’ (π β πΆ β π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrneq.1 | . 2 β’ (π β π΄ β π΅) | |
2 | tkgeom.p | . . . 4 β’ π = (BaseβπΊ) | |
3 | tkgeom.d | . . . 4 β’ β = (distβπΊ) | |
4 | tkgeom.i | . . . 4 β’ πΌ = (ItvβπΊ) | |
5 | tkgeom.g | . . . 4 β’ (π β πΊ β TarskiG) | |
6 | tgcgrcomlr.a | . . . 4 β’ (π β π΄ β π) | |
7 | tgcgrcomlr.b | . . . 4 β’ (π β π΅ β π) | |
8 | tgcgrcomlr.c | . . . 4 β’ (π β πΆ β π) | |
9 | tgcgrcomlr.d | . . . 4 β’ (π β π· β π) | |
10 | tgcgrcomlr.6 | . . . 4 β’ (π β (π΄ β π΅) = (πΆ β π·)) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | tgcgreqb 27770 | . . 3 β’ (π β (π΄ = π΅ β πΆ = π·)) |
12 | 11 | necon3bid 2985 | . 2 β’ (π β (π΄ β π΅ β πΆ β π·)) |
13 | 1, 12 | mpbid 231 | 1 β’ (π β πΆ β π·) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wne 2940 βcfv 6543 (class class class)co 7411 Basecbs 17146 distcds 17208 TarskiGcstrkg 27716 Itvcitv 27722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 df-trkgc 27737 df-trkg 27742 |
This theorem is referenced by: hlcgrex 27905 midexlem 27981 footexALT 28007 footexlem1 28008 footexlem2 28009 mideulem2 28023 opphllem3 28038 trgcopy 28093 iscgra1 28099 cgrane1 28101 cgrane2 28102 cgrcgra 28110 flatcgra 28113 cgrg3col4 28142 tgsas2 28145 tgsas3 28146 tgasa1 28147 |
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