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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 27732 | . . 3 β’ (π β (π΄ = π΅ β πΆ = π·)) |
12 | 11 | necon3bid 2986 | . 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 1542 β wcel 2107 β wne 2941 βcfv 6544 (class class class)co 7409 Basecbs 17144 distcds 17206 TarskiGcstrkg 27678 Itvcitv 27684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-trkgc 27699 df-trkg 27704 |
This theorem is referenced by: hlcgrex 27867 midexlem 27943 footexALT 27969 footexlem1 27970 footexlem2 27971 mideulem2 27985 opphllem3 28000 trgcopy 28055 iscgra1 28061 cgrane1 28063 cgrane2 28064 cgrcgra 28072 flatcgra 28075 cgrg3col4 28104 tgsas2 28107 tgsas3 28108 tgasa1 28109 |
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