Description: Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | tgcgrxfr.p | |- P = ( Base ` G ) |
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tgcgrxfr.m | |- .- = ( dist ` G ) |
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tgcgrxfr.i | |- I = ( Itv ` G ) |
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tgcgrxfr.r | |- .~ = ( cgrG ` G ) |
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tgcgrxfr.g | |- ( ph -> G e. TarskiG ) |
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tgbtwnxfr.a | |- ( ph -> A e. P ) |
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tgbtwnxfr.b | |- ( ph -> B e. P ) |
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tgbtwnxfr.c | |- ( ph -> C e. P ) |
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tgbtwnxfr.d | |- ( ph -> D e. P ) |
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tgbtwnxfr.e | |- ( ph -> E e. P ) |
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tgbtwnxfr.f | |- ( ph -> F e. P ) |
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tgbtwnxfr.2 | |- ( ph -> <" A B C "> .~ <" D E F "> ) |
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Assertion | cgr3rotr | |- ( ph -> <" C A B "> .~ <" F D E "> ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrxfr.p | |- P = ( Base ` G ) |
|
2 | tgcgrxfr.m | |- .- = ( dist ` G ) |
|
3 | tgcgrxfr.i | |- I = ( Itv ` G ) |
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4 | tgcgrxfr.r | |- .~ = ( cgrG ` G ) |
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5 | tgcgrxfr.g | |- ( ph -> G e. TarskiG ) |
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6 | tgbtwnxfr.a | |- ( ph -> A e. P ) |
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7 | tgbtwnxfr.b | |- ( ph -> B e. P ) |
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8 | tgbtwnxfr.c | |- ( ph -> C e. P ) |
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9 | tgbtwnxfr.d | |- ( ph -> D e. P ) |
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10 | tgbtwnxfr.e | |- ( ph -> E e. P ) |
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11 | tgbtwnxfr.f | |- ( ph -> F e. P ) |
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12 | tgbtwnxfr.2 | |- ( ph -> <" A B C "> .~ <" D E F "> ) |
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13 | 1 2 3 4 5 6 7 8 9 10 11 12 | cgr3swap23 | |- ( ph -> <" A C B "> .~ <" D F E "> ) |
14 | 1 2 3 4 5 6 8 7 9 11 10 13 | cgr3swap12 | |- ( ph -> <" C A B "> .~ <" F D E "> ) |