Theorem dalem56

Description: Lemma for dath . Analogue of dalem55 for line S T . (Contributed by NM, 8-Aug-2012)

	Ref	Expression
Hypotheses	dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem54.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem54.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem54.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem54.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem54.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem54.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem54.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
	dalem54.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
Assertion	dalem56	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B$$

Proof

Step	Hyp	Ref	Expression
1		dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
2		dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
6		dalem54.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem54.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem54.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem54.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem54.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11		dalem54.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
12		dalem54.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
13		dalem54.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
14	1 2 3 4	dalemswapyz	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
15	14	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
16		simp2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y=Z$$
17	16	eqcomd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Z=Y$$
18	1 2 3 4 5	dalemswapyzps	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(d\in A\wedge c\in A\right)\wedge \neg d\underset{\dot{}}{\le }Z\wedge \left(c\ne d\wedge \neg c\underset{\dot{}}{\le }Z\wedge C\underset{\dot{}}{\le }\left(d\underset{\dot{}}{\vee }c\right)\right)\right)$$
19		biid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)\leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
20		biid	$$⊢\left(\left(d\in A\wedge c\in A\right)\wedge \neg d\underset{\dot{}}{\le }Z\wedge \left(c\ne d\wedge \neg c\underset{\dot{}}{\le }Z\wedge C\underset{\dot{}}{\le }\left(d\underset{\dot{}}{\vee }c\right)\right)\right)\leftrightarrow \left(\left(d\in A\wedge c\in A\right)\wedge \neg d\underset{\dot{}}{\le }Z\wedge \left(c\ne d\wedge \neg c\underset{\dot{}}{\le }Z\wedge C\underset{\dot{}}{\le }\left(d\underset{\dot{}}{\vee }c\right)\right)\right)$$
21		eqid	$$⊢\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)=\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)$$
22		eqid	$$⊢\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)=\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)$$
23		eqid	$$⊢\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)=\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)$$
24		eqid	$$⊢\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z=\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z$$
25	19 2 3 4 20 6 7 9 8 21 22 23 24	dalem55	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)\wedge Z=Y\wedge \left(\left(d\in A\wedge c\in A\right)\wedge \neg d\underset{\dot{}}{\le }Z\wedge \left(c\ne d\wedge \neg c\underset{\dot{}}{\le }Z\wedge C\underset{\dot{}}{\le }\left(d\underset{\dot{}}{\vee }c\right)\right)\right)\right)\to \left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\wedge }\left(\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z\right)$$
26	15 17 18 25	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\wedge }\left(\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z\right)$$
27	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
28	27	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
29	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
30	29	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
31	5	dalemccea	$$⊢\psi \to c\in A$$
32	31	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in A$$
33	1	dalempea	$$⊢\phi \to P\in A$$
34	33	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\in A$$
35		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
36	35 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge c\in A\wedge P\in A\right)\to c\underset{\dot{}}{\vee }P\in {\mathrm{Base}}_K$$
37	30 32 34 36	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\vee }P\in {\mathrm{Base}}_K$$
38	5	dalemddea	$$⊢\psi \to d\in A$$
39	38	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\in A$$
40	1	dalemsea	$$⊢\phi \to S\in A$$
41	40	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to S\in A$$
42	35 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge d\in A\wedge S\in A\right)\to d\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
43	30 39 41 42	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
44	35 6	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge c\underset{\dot{}}{\vee }P\in {\mathrm{Base}}_K\wedge d\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)=\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)$$
45	28 37 43 44	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)=\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)$$
46	10 45	eqtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G=\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)$$
47	1	dalemqea	$$⊢\phi \to Q\in A$$
48	47	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Q\in A$$
49	35 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge c\in A\wedge Q\in A\right)\to c\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
50	30 32 48 49	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
51	1	dalemtea	$$⊢\phi \to T\in A$$
52	51	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to T\in A$$
53	35 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge d\in A\wedge T\in A\right)\to d\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
54	30 39 52 53	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
55	35 6	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge c\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge d\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\to \left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)=\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)$$
56	28 50 54 55	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)=\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)$$
57	11 56	eqtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H=\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)$$
58	46 57	oveq12d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H=\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)$$
59	58	oveq1d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
60	1	dalemrea	$$⊢\phi \to R\in A$$
61	60	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to R\in A$$
62	35 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge c\in A\wedge R\in A\right)\to c\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
63	30 32 61 62	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
64	1	dalemuea	$$⊢\phi \to U\in A$$
65	64	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to U\in A$$
66	35 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge d\in A\wedge U\in A\right)\to d\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
67	30 39 65 66	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
68	35 6	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge c\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K\wedge d\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K\right)\to \left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)=\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)$$
69	28 63 67 68	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)=\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)$$
70	12 69	eqtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I=\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)$$
71	58 70	oveq12d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I=\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)$$
72	71 16	oveq12d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y=\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z$$
73	13 72	eqtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B=\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z$$
74	58 73	oveq12d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B=\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\wedge }\left(\left(\left(\left(\left(d\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }P\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }Q\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(d\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(c\underset{\dot{}}{\vee }R\right)\right)\right)\underset{\dot{}}{\wedge }Z\right)$$
75	26 59 74	3eqtr4d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B$$