Theorem cdlemkuel-2N

Description: Part of proof of Lemma K of Crawley p. 118. Conditions for the sigma_2 (p) function to be a translation. TODO: combine cdlemkj ? (Contributed by NM, 2-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk2.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemk2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemk2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemk2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemk2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemk2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemk2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	cdlemk2.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
	cdlemk2.s	$$⊢S=\left(f\in T⟼\left(\iota i\in T|i\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(f\right)\right)\underset{\dot{}}{\wedge }\left(N\left(P\right)\underset{\dot{}}{\vee }R\left(f\circ F^{-1}\right)\right)\right)\right)$$
	cdlemk2.q	$$⊢Q=S\left(C\right)$$
	cdlemk2.v	$$⊢V=\left(d\in T⟼\left(\iota k\in T|k\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(d\right)\right)\underset{\dot{}}{\wedge }\left(Q\left(P\right)\underset{\dot{}}{\vee }R\left(d\circ C^{-1}\right)\right)\right)\right)$$
Assertion	cdlemkuel-2N	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge R\left(F\right)=R\left(N\right)\wedge G\in T\right)\wedge \left(F\in T\wedge C\in T\wedge N\in T\right)\wedge \left(\left(R\left(C\right)\ne R\left(F\right)\wedge R\left(C\right)\ne R\left(G\right)\right)\wedge \left(F\ne {\mathrm{I}\upharpoonright }_B\wedge G\ne {\mathrm{I}\upharpoonright }_B\wedge C\ne {\mathrm{I}\upharpoonright }_B\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\right)\to V\left(G\right)\in T$$

Proof

Step	Hyp	Ref	Expression
1		cdlemk2.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemk2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemk2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemk2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemk2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemk2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemk2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
8		cdlemk2.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
9		cdlemk2.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		cdlemk2.q	$$⊢Q=S\left(C\right)$$
11		cdlemk2.v	$$⊢V=\left(d\in T⟼\left(\iota k\in T|k\left(P\right)=\left(P\underset{\dot{}}{\vee }R\left(d\right)\right)\underset{\dot{}}{\wedge }\left(Q\left(P\right)\underset{\dot{}}{\vee }R\left(d\circ C^{-1}\right)\right)\right)\right)$$
12	1 2 3 4 5 6 7 8 9 10 11	cdlemkuel	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge R\left(F\right)=R\left(N\right)\wedge G\in T\right)\wedge \left(F\in T\wedge C\in T\wedge N\in T\right)\wedge \left(\left(R\left(C\right)\ne R\left(F\right)\wedge R\left(C\right)\ne R\left(G\right)\right)\wedge \left(F\ne {\mathrm{I}\upharpoonright }_B\wedge G\ne {\mathrm{I}\upharpoonright }_B\wedge C\ne {\mathrm{I}\upharpoonright }_B\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\right)\to V\left(G\right)\in T$$