Theorem cdleme50rnlem

Description: Part of proof of Lemma D in Crawley p. 113. TODO: fix comment. TODO: can we get rid of G stuff if we show G =`' F ` earlier? (Contributed by NM, 9-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef50.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemef50.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemef50.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemef50.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemef50.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemef50.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemef50.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdlemef50.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemefs50.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemef50.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
	cdlemef50.v	$$⊢V=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
	cdlemef50.n	$$⊢N=\left(v\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemefs50.o	$$⊢O=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(N\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemef50.g	$$⊢G=\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)$$
Assertion	cdleme50rnlem	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to \mathrm{ran}F=B$$

Proof

Step	Hyp	Ref	Expression
1		cdlemef50.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemef50.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemef50.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemef50.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemef50.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemef50.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemef50.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdlemef50.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdlemefs50.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdlemef50.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
11		cdlemef50.v	$$⊢V=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
12		cdlemef50.n	$$⊢N=\left(v\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
13		cdlemefs50.o	$$⊢O=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(N\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		cdlemef50.g	$$⊢G=\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)$$
15	1 2 3 4 5 6 7 8 9 10	cdleme50f	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to F:B⟶B$$
16	15	frnd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to \mathrm{ran}F\subseteq B$$
17	1 2 3 4 5 6 11 12 13 14	cdlemeg46fvcl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge e\in B\right)\to G\left(e\right)\in B$$
18	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme48fgv	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge e\in B\right)\to F\left(G\left(e\right)\right)=e$$
19		fveqeq2	$$⊢d=G\left(e\right)\to \left(F\left(d\right)=e\leftrightarrow F\left(G\left(e\right)\right)=e\right)$$
20	19	rspcev	$$⊢\left(G\left(e\right)\in B\wedge F\left(G\left(e\right)\right)=e\right)\to \exists d\in BF\left(d\right)=e$$
21	17 18 20	syl2anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge e\in B\right)\to \exists d\in BF\left(d\right)=e$$
22	15	adantr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge e\in B\right)\to F:B⟶B$$
23		ffn	$$⊢F:B⟶B\to FFnB$$
24		fvelrnb	$$⊢FFnB\to \left(e\in \mathrm{ran}F\leftrightarrow \exists d\in BF\left(d\right)=e\right)$$
25	22 23 24	3syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge e\in B\right)\to \left(e\in \mathrm{ran}F\leftrightarrow \exists d\in BF\left(d\right)=e\right)$$
26	21 25	mpbird	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge e\in B\right)\to e\in \mathrm{ran}F$$
27	16 26	eqelssd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to \mathrm{ran}F=B$$