Theorem cdlemn

Description: Lemma N of Crawley p. 121 line 27. (Contributed by NM, 27-Feb-2014)

	Ref	Expression
Hypotheses	cdlemn11.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemn11.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemn11.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemn11.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemn11.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemn11.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
	cdlemn11.J	$$⊢J=\mathrm{DIsoC}\left(K\right)\left(W\right)$$
	cdlemn11.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	cdlemn11.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
Assertion	cdlemn	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\leftrightarrow J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdlemn11.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemn11.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemn11.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemn11.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdlemn11.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdlemn11.i	$$⊢I=\mathrm{DIsoB}\left(K\right)\left(W\right)$$
7		cdlemn11.J	$$⊢J=\mathrm{DIsoC}\left(K\right)\left(W\right)$$
8		cdlemn11.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
9		cdlemn11.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(U\right)$$
10	1 2 3 4 5 8 9 6 7	cdlemn5	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\wedge R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\right)\to J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)$$
11	10	3expia	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\to J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)\right)$$
12	1 2 3 4 5 6 7 8 9	cdlemn11	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\wedge J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)\right)\to R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)$$
13	12	3expia	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\right)\to \left(J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)\to R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\right)$$
14	11 13	impbid	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }X\right)\leftrightarrow J\left(R\right)\subseteq J\left(Q\right)\underset{\dot{}}{\oplus }I\left(X\right)\right)$$