Theorem mdslmd1lem2 11898

Description: Lemma for mdslmd1i 11901.

Hypotheses

Ref	Expression
mdslmd.1	|- A e. CH
mdslmd.2	|- B e. CH
mdslmd.3	|- C e. CH
mdslmd.4	|- D e. CH
mdslmd1lem.5	|- R e. CH

Assertion

Ref	Expression
mdslmd1lem2	|- (((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) -> (((R i^i B) C_ (D i^i B) -> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B)))) -> (((C i^i D) C_ R /\ R C_ D) -> ((R vH C) i^i D) C_ (R vH (C i^i D)))))

Proof of Theorem mdslmd1lem2

Step	Hyp	Ref	Expression
1		simpllr 453	. . . . . 6 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> B MH* A)
2		mdslmd.3	. . . . . . . . . . . . . 14 |- C e. CH
3		mdslmd1lem.5	. . . . . . . . . . . . . 14 |- R e. CH
4	2, 3	chub2i 11026	. . . . . . . . . . . . 13 |- C C_ (R vH C)
5		sstr 2625	. . . . . . . . . . . . 13 |- ((A C_ C /\ C C_ (R vH C)) -> A C_ (R vH C))
6	4, 5	mpan2 760	. . . . . . . . . . . 12 |- (A C_ C -> A C_ (R vH C))
7	6	ad2antrr 440	. . . . . . . . . . 11 |- (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) -> A C_ (R vH C))
8	7	ad2antlr 441	. . . . . . . . . 10 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ (R vH C))
9		simplr 449	. . . . . . . . . . 11 |- (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) -> A C_ D)
10	9	ad2antlr 441	. . . . . . . . . 10 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ D)
11	8, 10	jca 310	. . . . . . . . 9 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (A C_ (R vH C) /\ A C_ D))
12		ssin 2814	. . . . . . . . 9 |- ((A C_ (R vH C) /\ A C_ D) <-> A C_ ((R vH C) i^i D))
13	11, 12	sylib 215	. . . . . . . 8 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ ((R vH C) i^i D))
14		ssin 2814	. . . . . . . . . . 11 |- ((A C_ C /\ A C_ D) <-> A C_ (C i^i D))
15		mdslmd.4	. . . . . . . . . . . . . 14 |- D e. CH
16	2, 15	chincli 11016	. . . . . . . . . . . . 13 |- (C i^i D) e. CH
17	16, 3	chub2i 11026	. . . . . . . . . . . 12 |- (C i^i D) C_ (R vH (C i^i D))
18		sstr 2625	. . . . . . . . . . . 12 |- ((A C_ (C i^i D) /\ (C i^i D) C_ (R vH (C i^i D))) -> A C_ (R vH (C i^i D)))
19	17, 18	mpan2 760	. . . . . . . . . . 11 |- (A C_ (C i^i D) -> A C_ (R vH (C i^i D)))
20	14, 19	sylbi 216	. . . . . . . . . 10 |- ((A C_ C /\ A C_ D) -> A C_ (R vH (C i^i D)))
21	20	adantr 425	. . . . . . . . 9 |- (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) -> A C_ (R vH (C i^i D)))
22	21	ad2antlr 441	. . . . . . . 8 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ (R vH (C i^i D)))
23	13, 22	jca 310	. . . . . . 7 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (A C_ ((R vH C) i^i D) /\ A C_ (R vH (C i^i D))))
24		ssin 2814	. . . . . . 7 |- ((A C_ ((R vH C) i^i D) /\ A C_ (R vH (C i^i D))) <-> A C_ (((R vH C) i^i D) i^i (R vH (C i^i D))))
25	23, 24	sylib 215	. . . . . 6 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ (((R vH C) i^i D) i^i (R vH (C i^i D))))
26		inss2 2813	. . . . . . . . . . 11 |- ((R vH C) i^i D) C_ D
27		sstr 2625	. . . . . . . . . . 11 |- ((((R vH C) i^i D) C_ D /\ D C_ (A vH B)) -> ((R vH C) i^i D) C_ (A vH B))
28	26, 27	mpan 759	. . . . . . . . . 10 |- (D C_ (A vH B) -> ((R vH C) i^i D) C_ (A vH B))
29	28	ad2antll 443	. . . . . . . . 9 |- (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) -> ((R vH C) i^i D) C_ (A vH B))
30	29	ad2antlr 441	. . . . . . . 8 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((R vH C) i^i D) C_ (A vH B))
31		sstr 2625	. . . . . . . . . . . . . 14 |- ((R C_ D /\ D C_ (A vH B)) -> R C_ (A vH B))
32	31	ancoms 484	. . . . . . . . . . . . 13 |- ((D C_ (A vH B) /\ R C_ D) -> R C_ (A vH B))
33	32	ad2ant2l 444	. . . . . . . . . . . 12 |- (((C C_ (A vH B) /\ D C_ (A vH B)) /\ ((C i^i D) C_ R /\ R C_ D)) -> R C_ (A vH B))
34	33	adantll 428	. . . . . . . . . . 11 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> R C_ (A vH B))
35	34	adantll 428	. . . . . . . . . 10 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> R C_ (A vH B))
36		ssinss1 2821	. . . . . . . . . . . 12 |- (C C_ (A vH B) -> (C i^i D) C_ (A vH B))
37	36	ad2antrl 442	. . . . . . . . . . 11 |- (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) -> (C i^i D) C_ (A vH B))
38	37	ad2antlr 441	. . . . . . . . . 10 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (C i^i D) C_ (A vH B))
39	35, 38	jca 310	. . . . . . . . 9 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (R C_ (A vH B) /\ (C i^i D) C_ (A vH B)))
40		mdslmd.1	. . . . . . . . . . 11 |- A e. CH
41		mdslmd.2	. . . . . . . . . . 11 |- B e. CH
42	40, 41	chjcli 11013	. . . . . . . . . 10 |- (A vH B) e. CH
43	3, 16, 42	chlubi 11027	. . . . . . . . 9 |- ((R C_ (A vH B) /\ (C i^i D) C_ (A vH B)) <-> (R vH (C i^i D)) C_ (A vH B))
44	39, 43	sylib 215	. . . . . . . 8 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (R vH (C i^i D)) C_ (A vH B))
45	30, 44	jca 310	. . . . . . 7 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (((R vH C) i^i D) C_ (A vH B) /\ (R vH (C i^i D)) C_ (A vH B)))
46	3, 2	chjcli 11013	. . . . . . . . 9 |- (R vH C) e. CH
47	46, 15	chincli 11016	. . . . . . . 8 |- ((R vH C) i^i D) e. CH
48	3, 16	chjcli 11013	. . . . . . . 8 |- (R vH (C i^i D)) e. CH
49	47, 48, 42	chlubi 11027	. . . . . . 7 |- ((((R vH C) i^i D) C_ (A vH B) /\ (R vH (C i^i D)) C_ (A vH B)) <-> (((R vH C) i^i D) vH (R vH (C i^i D))) C_ (A vH B))
50	45, 49	sylib 215	. . . . . 6 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (((R vH C) i^i D) vH (R vH (C i^i D))) C_ (A vH B))
51	40, 41, 47, 48	mdslle1i 11889	. . . . . 6 |- ((B MH* A /\ A C_ (((R vH C) i^i D) i^i (R vH (C i^i D))) /\ (((R vH C) i^i D) vH (R vH (C i^i D))) C_ (A vH B)) -> (((R vH C) i^i D) C_ (R vH (C i^i D)) <-> (((R vH C) i^i D) i^i B) C_ ((R vH (C i^i D)) i^i B)))
52	1, 25, 50, 51	syl111anc 1100	. . . . 5 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (((R vH C) i^i D) C_ (R vH (C i^i D)) <-> (((R vH C) i^i D) i^i B) C_ ((R vH (C i^i D)) i^i B)))
53	40, 41, 3, 2	mdslj1i 11891	. . . . . . . . . 10 |- (((A MH B /\ B MH* A) /\ (A C_ (R i^i C) /\ (R vH C) C_ (A vH B))) -> ((R vH C) i^i B) = ((R i^i B) vH (C i^i B)))
54		sstr 2625	. . . . . . . . . . . . . . 15 |- ((A C_ (C i^i D) /\ (C i^i D) C_ R) -> A C_ R)
55	54, 14	sylanb 498	. . . . . . . . . . . . . 14 |- (((A C_ C /\ A C_ D) /\ (C i^i D) C_ R) -> A C_ R)
56	55	ad2ant2r 445	. . . . . . . . . . . . 13 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ R)
57		simplll 452	. . . . . . . . . . . . 13 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ C)
58	56, 57	jca 310	. . . . . . . . . . . 12 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (A C_ R /\ A C_ C))
59		ssin 2814	. . . . . . . . . . . 12 |- ((A C_ R /\ A C_ C) <-> A C_ (R i^i C))
60	58, 59	sylib 215	. . . . . . . . . . 11 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> A C_ (R i^i C))
61		simplrl 454	. . . . . . . . . . . . 13 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> C C_ (A vH B))
62	34, 61	jca 310	. . . . . . . . . . . 12 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (R C_ (A vH B) /\ C C_ (A vH B)))
63	3, 2, 42	chlubi 11027	. . . . . . . . . . . 12 |- ((R C_ (A vH B) /\ C C_ (A vH B)) <-> (R vH C) C_ (A vH B))
64	62, 63	sylib 215	. . . . . . . . . . 11 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (R vH C) C_ (A vH B))
65	60, 64	jca 310	. . . . . . . . . 10 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (A C_ (R i^i C) /\ (R vH C) C_ (A vH B)))
66	53, 65	sylan2 500	. . . . . . . . 9 |- (((A MH B /\ B MH* A) /\ (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D))) -> ((R vH C) i^i B) = ((R i^i B) vH (C i^i B)))
67	66	anassrs 489	. . . . . . . 8 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((R vH C) i^i B) = ((R i^i B) vH (C i^i B)))
68	67	ineq1d 2795	. . . . . . 7 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (((R vH C) i^i B) i^i (D i^i B)) = (((R i^i B) vH (C i^i B)) i^i (D i^i B)))
69		inindir 2810	. . . . . . 7 |- (((R vH C) i^i D) i^i B) = (((R vH C) i^i B) i^i (D i^i B))
70	68, 69	syl5req 1941	. . . . . 6 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) = (((R vH C) i^i D) i^i B))
71	40, 41, 3, 16	mdslj1i 11891	. . . . . . . . 9 |- (((A MH B /\ B MH* A) /\ (A C_ (R i^i (C i^i D)) /\ (R vH (C i^i D)) C_ (A vH B))) -> ((R vH (C i^i D)) i^i B) = ((R i^i B) vH ((C i^i D) i^i B)))
72	14	biimpi 168	. . . . . . . . . . . . . 14 |- ((A C_ C /\ A C_ D) -> A C_ (C i^i D))
73	72	adantr 425	. . . . . . . . . . . . 13 |- (((A C_ C /\ A C_ D) /\ (C i^i D) C_ R) -> A C_ (C i^i D))
74	55, 73	jca 310	. . . . . . . . . . . 12 |- (((A C_ C /\ A C_ D) /\ (C i^i D) C_ R) -> (A C_ R /\ A C_ (C i^i D)))
75		ssin 2814	. . . . . . . . . . . 12 |- ((A C_ R /\ A C_ (C i^i D)) <-> A C_ (R i^i (C i^i D)))
76	74, 75	sylib 215	. . . . . . . . . . 11 |- (((A C_ C /\ A C_ D) /\ (C i^i D) C_ R) -> A C_ (R i^i (C i^i D)))
77	32	adantll 428	. . . . . . . . . . . . 13 |- (((C C_ (A vH B) /\ D C_ (A vH B)) /\ R C_ D) -> R C_ (A vH B))
78	36	ad2antrr 440	. . . . . . . . . . . . 13 |- (((C C_ (A vH B) /\ D C_ (A vH B)) /\ R C_ D) -> (C i^i D) C_ (A vH B))
79	77, 78	jca 310	. . . . . . . . . . . 12 |- (((C C_ (A vH B) /\ D C_ (A vH B)) /\ R C_ D) -> (R C_ (A vH B) /\ (C i^i D) C_ (A vH B)))
80	79, 43	sylib 215	. . . . . . . . . . 11 |- (((C C_ (A vH B) /\ D C_ (A vH B)) /\ R C_ D) -> (R vH (C i^i D)) C_ (A vH B))
81	76, 80	anim12i 360	. . . . . . . . . 10 |- ((((A C_ C /\ A C_ D) /\ (C i^i D) C_ R) /\ ((C C_ (A vH B) /\ D C_ (A vH B)) /\ R C_ D)) -> (A C_ (R i^i (C i^i D)) /\ (R vH (C i^i D)) C_ (A vH B)))
82	81	an4s 566	. . . . . . . . 9 |- ((((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (A C_ (R i^i (C i^i D)) /\ (R vH (C i^i D)) C_ (A vH B)))
83	71, 82	sylan2 500	. . . . . . . 8 |- (((A MH B /\ B MH* A) /\ (((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B))) /\ ((C i^i D) C_ R /\ R C_ D))) -> ((R vH (C i^i D)) i^i B) = ((R i^i B) vH ((C i^i D) i^i B)))
84	83	anassrs 489	. . . . . . 7 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((R vH (C i^i D)) i^i B) = ((R i^i B) vH ((C i^i D) i^i B)))
85		inindir 2810	. . . . . . . . 9 |- ((C i^i D) i^i B) = ((C i^i B) i^i (D i^i B))
86	85	a1i 8	. . . . . . . 8 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((C i^i D) i^i B) = ((C i^i B) i^i (D i^i B)))
87	86	opreq2d 4898	. . . . . . 7 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((R i^i B) vH ((C i^i D) i^i B)) = ((R i^i B) vH ((C i^i B) i^i (D i^i B))))
88	84, 87	eqtr2d 1926	. . . . . 6 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((R i^i B) vH ((C i^i B) i^i (D i^i B))) = ((R vH (C i^i D)) i^i B))
89	70, 88	sseq12d 2646	. . . . 5 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> ((((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B))) <-> (((R vH C) i^i D) i^i B) C_ ((R vH (C i^i D)) i^i B)))
90	52, 89	bitr4d 590	. . . 4 |- ((((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) /\ ((C i^i D) C_ R /\ R C_ D)) -> (((R vH C) i^i D) C_ (R vH (C i^i D)) <-> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B)))))
91	90	exbiri 421	. . 3 |- (((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) -> (((C i^i D) C_ R /\ R C_ D) -> ((((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B))) -> ((R vH C) i^i D) C_ (R vH (C i^i D)))))
92	91	a2d 16	. 2 |- (((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) -> ((((C i^i D) C_ R /\ R C_ D) -> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B)))) -> (((C i^i D) C_ R /\ R C_ D) -> ((R vH C) i^i D) C_ (R vH (C i^i D)))))
93		ssrin 2817	. . . 4 |- (R C_ D -> (R i^i B) C_ (D i^i B))
94	93	adantl 424	. . 3 |- (((C i^i D) C_ R /\ R C_ D) -> (R i^i B) C_ (D i^i B))
95	94	imim1i 19	. 2 |- (((R i^i B) C_ (D i^i B) -> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B)))) -> (((C i^i D) C_ R /\ R C_ D) -> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B)))))
96	92, 95	syl5 20	1 |- (((A MH B /\ B MH* A) /\ ((A C_ C /\ A C_ D) /\ (C C_ (A vH B) /\ D C_ (A vH B)))) -> (((R i^i B) C_ (D i^i B) -> (((R i^i B) vH (C i^i B)) i^i (D i^i B)) C_ ((R i^i B) vH ((C i^i B) i^i (D i^i B)))) -> (((C i^i D) C_ R /\ R C_ D) -> ((R vH C) i^i D) C_ (R vH (C i^i D)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   i^i cin 2592   C_ wss 2593   class class class wbr 3338  (class class class)co 4884  CHcch 10430   vH chj 10434   MH cmd 10467   MH* cdmd 10468

This theorem is referenced by:  mdslmd1lem4 11900

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585  ax-hcompl 10704

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-bases 8863  df-topgen 8864  df-cld 8939  df-ntr 8940  df-cls 8941  df-cn 9030  df-cnp 9031  df-haus 9059  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-ip 9689  df-ph 9813  df-hnorm 10469  df-hvsub 10472  df-hlim 10473  df-hcau 10474  df-sh 10709  df-ch 10725  df-oc 10757  df-ch0 10758  df-shsum 10906  df-chj 10908  df-md 11852  df-dmd 11853

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