Theorem cvlsupr4

Description: Consequence of superposition condition ( P .\/ R ) = ( Q .\/ R ) . (Contributed by NM, 9-Nov-2012)

	Ref	Expression
Hypotheses	cvlsupr2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cvlsupr2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cvlsupr2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
Assertion	cvlsupr4	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		cvlsupr2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		cvlsupr2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cvlsupr2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4	1 2 3	cvlsupr2	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\leftrightarrow \left(R\ne P\wedge R\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
5		simp3	$$⊢\left(R\ne P\wedge R\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
6	4 5	biimtrdi	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
7	6	3exp	$$⊢K\in \mathrm{CvLat}\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(P\ne Q\to \left(P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)$$
8	7	imp4a	$$⊢K\in \mathrm{CvLat}\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(\left(P\ne Q\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
9	8	3imp	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$