Theorem meetval2lem

Description: Lemma for meetval2 and meeteu . (Contributed by NM, 12-Sep-2018) TODO: combine this through meeteu into meetlem ?

	Ref	Expression
Hypotheses	meetval2.b	$$⊢B={\mathrm{Base}}_K$$
	meetval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	meetval2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	meetval2.k	$$⊢\phi \to K\in V$$
	meetval2.x	$$⊢\phi \to X\in B$$
	meetval2.y	$$⊢\phi \to Y\in B$$
Assertion	meetval2lem	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(\left(\forall y\in \left\{X,Y\right\}x\underset{\dot{}}{\le }y\wedge \forall z\in B\left(\forall y\in \left\{X,Y\right\}z\underset{\dot{}}{\le }y\to z\underset{\dot{}}{\le }x\right)\right)\leftrightarrow \left(\left(x\underset{\dot{}}{\le }X\wedge x\underset{\dot{}}{\le }Y\right)\wedge \forall z\in B\left(\left(z\underset{\dot{}}{\le }X\wedge z\underset{\dot{}}{\le }Y\right)\to z\underset{\dot{}}{\le }x\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		meetval2.b	$$⊢B={\mathrm{Base}}_K$$
2		meetval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		meetval2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		meetval2.k	$$⊢\phi \to K\in V$$
5		meetval2.x	$$⊢\phi \to X\in B$$
6		meetval2.y	$$⊢\phi \to Y\in B$$
7		breq2	$$⊢y=X\to \left(x\underset{\dot{}}{\le }y\leftrightarrow x\underset{\dot{}}{\le }X\right)$$
8		breq2	$$⊢y=Y\to \left(x\underset{\dot{}}{\le }y\leftrightarrow x\underset{\dot{}}{\le }Y\right)$$
9	7 8	ralprg	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(\forall y\in \left\{X,Y\right\}x\underset{\dot{}}{\le }y\leftrightarrow \left(x\underset{\dot{}}{\le }X\wedge x\underset{\dot{}}{\le }Y\right)\right)$$
10		breq2	$$⊢y=X\to \left(z\underset{\dot{}}{\le }y\leftrightarrow z\underset{\dot{}}{\le }X\right)$$
11		breq2	$$⊢y=Y\to \left(z\underset{\dot{}}{\le }y\leftrightarrow z\underset{\dot{}}{\le }Y\right)$$
12	10 11	ralprg	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(\forall y\in \left\{X,Y\right\}z\underset{\dot{}}{\le }y\leftrightarrow \left(z\underset{\dot{}}{\le }X\wedge z\underset{\dot{}}{\le }Y\right)\right)$$
13	12	imbi1d	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(\left(\forall y\in \left\{X,Y\right\}z\underset{\dot{}}{\le }y\to z\underset{\dot{}}{\le }x\right)\leftrightarrow \left(\left(z\underset{\dot{}}{\le }X\wedge z\underset{\dot{}}{\le }Y\right)\to z\underset{\dot{}}{\le }x\right)\right)$$
14	13	ralbidv	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(\forall z\in B\left(\forall y\in \left\{X,Y\right\}z\underset{\dot{}}{\le }y\to z\underset{\dot{}}{\le }x\right)\leftrightarrow \forall z\in B\left(\left(z\underset{\dot{}}{\le }X\wedge z\underset{\dot{}}{\le }Y\right)\to z\underset{\dot{}}{\le }x\right)\right)$$
15	9 14	anbi12d	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(\left(\forall y\in \left\{X,Y\right\}x\underset{\dot{}}{\le }y\wedge \forall z\in B\left(\forall y\in \left\{X,Y\right\}z\underset{\dot{}}{\le }y\to z\underset{\dot{}}{\le }x\right)\right)\leftrightarrow \left(\left(x\underset{\dot{}}{\le }X\wedge x\underset{\dot{}}{\le }Y\right)\wedge \forall z\in B\left(\left(z\underset{\dot{}}{\le }X\wedge z\underset{\dot{}}{\le }Y\right)\to z\underset{\dot{}}{\le }x\right)\right)\right)$$