Theorem cdlemkuvN

Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma_1 (p) function U . (Contributed by NM, 2-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk1.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemk1.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemk1.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemk1.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemk1.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemk1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemk1.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	cdlemk1.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
	cdlemk1.s	$$⊢S=\left(f\in T⟼\left(\iota i\in T|i\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(f\right)\right)\underset{\dot{}}{\wedge }\left(N\left(P\right)\underset{\dot{}}{\vee }R\left(f\circ F^{-1}\right)\right)\right)\right)$$
	cdlemk1.o	$$⊢O=S\left(D\right)$$
	cdlemk1.u	$$⊢U=\left(e\in T⟼\left(\iota j\in T|j\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(e\right)\right)\underset{\dot{}}{\wedge }\left(O\left(P\right)\underset{\dot{}}{\vee }R\left(e\circ D^{-1}\right)\right)\right)\right)$$
Assertion	cdlemkuvN	$$⊢G\in T\to U\left(G\right)=\left(\iota j\in T|j\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(G\right)\right)\underset{\dot{}}{\wedge }\left(O\left(P\right)\underset{\dot{}}{\vee }R\left(G\circ D^{-1}\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemk1.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemk1.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemk1.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemk1.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemk1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemk1.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
8		cdlemk1.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
9		cdlemk1.s	$$⊢S=\left(f\in T⟼\left(\iota i\in T|i\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(f\right)\right)\underset{\dot{}}{\wedge }\left(N\left(P\right)\underset{\dot{}}{\vee }R\left(f\circ F^{-1}\right)\right)\right)\right)$$
10		cdlemk1.o	$$⊢O=S\left(D\right)$$
11		cdlemk1.u	$$⊢U=\left(e\in T⟼\left(\iota j\in T|j\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(e\right)\right)\underset{\dot{}}{\wedge }\left(O\left(P\right)\underset{\dot{}}{\vee }R\left(e\circ D^{-1}\right)\right)\right)\right)$$
12	1 2 3 5 6 7 8 4 11	cdlemksv	$$⊢G\in T\to U\left(G\right)=\left(\iota j\in T|j\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(G\right)\right)\underset{\dot{}}{\wedge }\left(O\left(P\right)\underset{\dot{}}{\vee }R\left(G\circ D^{-1}\right)\right)\right)$$