Theorem dihord2

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. TODO: do we need -. X .<_ W and -. Y .<_ W ? (Contributed by NM, 4-Mar-2014)

	Ref	Expression
Hypotheses	dihord2.b	$$⊢B={\mathrm{Base}}_K$$
	dihord2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dihord2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dihord2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dihord2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dihord2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dihord2.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
	dihord2.J	$$⊢J=\mathrm{DIsoC}\left(K\right)\left(W\right)$$
	dihord2.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dihord2.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
Assertion	dihord2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(N\in A\wedge \neg N\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge \left(Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\wedge N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)=Y\wedge J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }W\right)\subseteq J\left(N\right)\underset{\dot{}}{\oplus }I\left(Y\underset{\dot{}}{\wedge }W\right)\right)\right)\to X\underset{\dot{}}{\le }Y$$

Proof

Step	Hyp	Ref	Expression
1		dihord2.b	$$⊢B={\mathrm{Base}}_K$$
2		dihord2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dihord2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dihord2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		dihord2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		dihord2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		dihord2.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
8		dihord2.J	$$⊢J=\mathrm{DIsoC}\left(K\right)\left(W\right)$$
9		dihord2.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
10		dihord2.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
11		eqid	$$⊢\mathrm{LTrn}\left(K\right)\left(W\right)=\mathrm{LTrn}\left(K\right)\left(W\right)$$
12		eqid	$$⊢\mathrm{trL}\left(K\right)\left(W\right)=\mathrm{trL}\left(K\right)\left(W\right)$$
13		eqid	$$⊢\left(h\in \mathrm{LTrn}\left(K\right)\left(W\right)⟼{\mathrm{I}\upharpoonright }_B\right)=\left(h\in \mathrm{LTrn}\left(K\right)\left(W\right)⟼{\mathrm{I}\upharpoonright }_B\right)$$
14		eqid	$$⊢\mathrm{oc}\left(K\right)\left(W\right)=\mathrm{oc}\left(K\right)\left(W\right)$$
15		eqid	$$⊢\mathrm{TEndo}\left(K\right)\left(W\right)=\mathrm{TEndo}\left(K\right)\left(W\right)$$
16		eqid	$$⊢{+}_U={+}_U$$
17		eqid	$$⊢\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=N\right)=\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=N\right)$$
18	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17	dihord2pre2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(N\in A\wedge \neg N\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge \left(Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\wedge N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)=Y\wedge J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }W\right)\subseteq J\left(N\right)\underset{\dot{}}{\oplus }I\left(Y\underset{\dot{}}{\wedge }W\right)\right)\right)\to \left(Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\le }\left(N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)\right)$$
19		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(N\in A\wedge \neg N\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge \left(Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\wedge N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)=Y\wedge J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }W\right)\subseteq J\left(N\right)\underset{\dot{}}{\oplus }I\left(Y\underset{\dot{}}{\wedge }W\right)\right)\right)\to Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X$$
20		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(N\in A\wedge \neg N\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge \left(Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\wedge N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)=Y\wedge J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }W\right)\subseteq J\left(N\right)\underset{\dot{}}{\oplus }I\left(Y\underset{\dot{}}{\wedge }W\right)\right)\right)\to N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)=Y$$
21	18 19 20	3brtr3d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(N\in A\wedge \neg N\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge \left(Q\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\wedge N\underset{\dot{}}{\vee }\left(Y\underset{\dot{}}{\wedge }W\right)=Y\wedge J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\underset{\dot{}}{\wedge }W\right)\subseteq J\left(N\right)\underset{\dot{}}{\oplus }I\left(Y\underset{\dot{}}{\wedge }W\right)\right)\right)\to X\underset{\dot{}}{\le }Y$$