Theorem cdlemn5

Description: Part of proof of Lemma N of Crawley p. 121 line 32. (Contributed by NM, 25-Feb-2014)

	Ref	Expression
Hypotheses	cdlemn5.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemn5.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemn5.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemn5.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemn5.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemn5.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	cdlemn5.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
	cdlemn5.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
	cdlemn5.J	$$⊢J=\mathrm{DIsoC}\left(K\right)\left(W\right)$$
Assertion	cdlemn5	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\wedge R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\right)\to J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdlemn5.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemn5.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemn5.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemn5.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdlemn5.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdlemn5.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
7		cdlemn5.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
8		cdlemn5.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
9		cdlemn5.J	$$⊢J=\mathrm{DIsoC}\left(K\right)\left(W\right)$$
10		eqid	$$⊢\mathrm{oc}\left(K\right)\left(W\right)=\mathrm{oc}\left(K\right)\left(W\right)$$
11		eqid	$$⊢\left(h\in \mathrm{LTrn}\left(K\right)\left(W\right)⟼{\mathrm{I}\upharpoonright }_B\right)=\left(h\in \mathrm{LTrn}\left(K\right)\left(W\right)⟼{\mathrm{I}\upharpoonright }_B\right)$$
12		eqid	$$⊢\mathrm{LTrn}\left(K\right)\left(W\right)=\mathrm{LTrn}\left(K\right)\left(W\right)$$
13		eqid	$$⊢\mathrm{TEndo}\left(K\right)\left(W\right)=\mathrm{TEndo}\left(K\right)\left(W\right)$$
14		eqid	$$⊢\mathrm{LSpan}\left(U\right)=\mathrm{LSpan}\left(U\right)$$
15		eqid	$$⊢\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=Q\right)=\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=Q\right)$$
16		eqid	$$⊢\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=R\right)=\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(\mathrm{oc}\left(K\right)\left(W\right)\right)=R\right)$$
17		eqid	$$⊢\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(Q\right)=R\right)=\left(\iota h\in \mathrm{LTrn}\left(K\right)\left(W\right)|h\left(Q\right)=R\right)$$
18	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17	cdlemn5pre	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\wedge R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\right)\to J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)$$