Theorem cdleme43aN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v_1). (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme43.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme43.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme43.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme43.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme43.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme43.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme43.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme43.x	$$⊢X=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
	cdleme43.c	$$⊢C=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme43.f	$$⊢Z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(C\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme43.d	$$⊢D=\left(S\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme43.g	$$⊢G=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme43.e	$$⊢E=\left(D\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme43.v	$$⊢V=\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme43.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme43aN	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }V\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme43.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme43.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme43.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme43.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme43.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme43.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme43.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme43.x	$$⊢X=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
9		cdleme43.c	$$⊢C=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme43.f	$$⊢Z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(C\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme43.d	$$⊢D=\left(S\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12		cdleme43.g	$$⊢G=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
13		cdleme43.e	$$⊢E=\left(D\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		cdleme43.v	$$⊢V=\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
15		cdleme43.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
16	3 5	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }P$$
17	14	oveq2i	$$⊢D\underset{\dot{}}{\vee }V=D\underset{\dot{}}{\vee }\left(\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)$$
18	17	a1i	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to D\underset{\dot{}}{\vee }V=D\underset{\dot{}}{\vee }\left(\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)$$
19	16 18	oveq12d	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }V\right)=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(Z\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
20	12 19	eqtr4id	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }V\right)$$