Metamath Proof Explorer


Theorem cdleme51finvN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemef50.b B = Base K
cdlemef50.l ˙ = K
cdlemef50.j ˙ = join K
cdlemef50.m ˙ = meet K
cdlemef50.a A = Atoms K
cdlemef50.h H = LHyp K
cdlemef50.u U = P ˙ Q ˙ W
cdlemef50.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
cdlemefs50.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
cdlemef50.f F = x B if P Q ¬ x ˙ W ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W x
cdlemef51.v V = Q ˙ P ˙ W
cdlemef51.n N = v ˙ V ˙ P ˙ Q ˙ v ˙ W
cdlemefs51.o O = Q ˙ P ˙ N ˙ u ˙ v ˙ W
cdlemef51.g G = a B if Q P ¬ a ˙ W ι c B | u A ¬ u ˙ W u ˙ a ˙ W = a c = if u ˙ Q ˙ P ι b B | v A ¬ v ˙ W ¬ v ˙ Q ˙ P b = O u / v N ˙ a ˙ W a
Assertion cdleme51finvN K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F -1 = G

Proof

Step Hyp Ref Expression
1 cdlemef50.b B = Base K
2 cdlemef50.l ˙ = K
3 cdlemef50.j ˙ = join K
4 cdlemef50.m ˙ = meet K
5 cdlemef50.a A = Atoms K
6 cdlemef50.h H = LHyp K
7 cdlemef50.u U = P ˙ Q ˙ W
8 cdlemef50.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
9 cdlemefs50.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
10 cdlemef50.f F = x B if P Q ¬ x ˙ W ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W x
11 cdlemef51.v V = Q ˙ P ˙ W
12 cdlemef51.n N = v ˙ V ˙ P ˙ Q ˙ v ˙ W
13 cdlemefs51.o O = Q ˙ P ˙ N ˙ u ˙ v ˙ W
14 cdlemef51.g G = a B if Q P ¬ a ˙ W ι c B | u A ¬ u ˙ W u ˙ a ˙ W = a c = if u ˙ Q ˙ P ι b B | v A ¬ v ˙ W ¬ v ˙ Q ˙ P b = O u / v N ˙ a ˙ W a
15 1 2 3 4 5 6 7 8 9 10 cdleme50f1o K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F : B 1-1 onto B
16 dff1o4 F : B 1-1 onto B F Fn B F -1 Fn B
17 15 16 sylib K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F Fn B F -1 Fn B
18 17 simprd K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F -1 Fn B
19 1 2 3 4 5 6 11 12 13 14 cdleme50f1o K HL W H Q A ¬ Q ˙ W P A ¬ P ˙ W G : B 1-1 onto B
20 19 3com23 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W G : B 1-1 onto B
21 f1ofn G : B 1-1 onto B G Fn B
22 20 21 syl K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W G Fn B
23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme51finvfvN K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W e B F -1 e = G e
24 18 22 23 eqfnfvd K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W F -1 = G