Theorem mdslmd1lem2 29185

Description: Lemma for mdslmd1i 29188. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

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		ssrin 3838	. . . 4 |- ( R C_ D -> ( R i^i B ) C_ ( D i^i B ) )
2	1	adantl 482	. . 3 |- ( ( ( C i^i D ) C_ R /\ R C_ D ) -> ( R i^i B ) C_ ( D i^i B ) )
3	2	imim1i 63	. 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 ) ) ) ) )
4		simpllr 799	. . . . . 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 )
5		mdslmd.3	. . . . . . . . . . . 12 |- C e. CH
6		mdslmd1lem.5	. . . . . . . . . . . 12 |- R e. CH
7	5, 6	chub2i 28329	. . . . . . . . . . 11 |- C C_ ( R vH C )
8		sstr 3611	. . . . . . . . . . 11 |- ( ( A C_ C /\ C C_ ( R vH C ) ) -> A C_ ( R vH C ) )
9	7, 8	mpan2 707	. . . . . . . . . 10 |- ( A C_ C -> A C_ ( R vH C ) )
10	9	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A C_ C /\ A C_ D ) /\ ( C C_ ( A vH B ) /\ D C_ ( A vH B ) ) ) -> A C_ ( R vH C ) )
11	10	ad2antlr 763	. . . . . . . 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 ) )
12		simplr 792	. . . . . . . . 9 |- ( ( ( A C_ C /\ A C_ D ) /\ ( C C_ ( A vH B ) /\ D C_ ( A vH B ) ) ) -> A C_ D )
13	12	ad2antlr 763	. . . . . . . 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_ D )
14	11, 13	ssind 3837	. . . . . . 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 ) )
15		ssin 3835	. . . . . . . . . 10 |- ( ( A C_ C /\ A C_ D ) <-> A C_ ( C i^i D ) )
16		mdslmd.4	. . . . . . . . . . . . 13 |- D e. CH
17	5, 16	chincli 28319	. . . . . . . . . . . 12 |- ( C i^i D ) e. CH
18	17, 6	chub2i 28329	. . . . . . . . . . 11 |- ( C i^i D ) C_ ( R vH ( C i^i D ) )
19		sstr 3611	. . . . . . . . . . 11 |- ( ( 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 ) ) )
20	18, 19	mpan2 707	. . . . . . . . . 10 |- ( A C_ ( C i^i D ) -> A C_ ( R vH ( C i^i D ) ) )
21	15, 20	sylbi 207	. . . . . . . . 9 |- ( ( A C_ C /\ A C_ D ) -> A C_ ( R vH ( C i^i D ) ) )
22	21	adantr 481	. . . . . . . 8 |- ( ( ( A C_ C /\ A C_ D ) /\ ( C C_ ( A vH B ) /\ D C_ ( A vH B ) ) ) -> A C_ ( R vH ( C i^i D ) ) )
23	22	ad2antlr 763	. . . . . . 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 ) ) )
24	14, 23	ssind 3837	. . . . . 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 ) ) ) )
25		inss2 3834	. . . . . . . . . . 11 |- ( ( R vH C ) i^i D ) C_ D
26		sstr 3611	. . . . . . . . . . 11 |- ( ( ( ( R vH C ) i^i D ) C_ D /\ D C_ ( A vH B ) ) -> ( ( R vH C ) i^i D ) C_ ( A vH B ) )
27	25, 26	mpan 706	. . . . . . . . . 10 |- ( D C_ ( A vH B ) -> ( ( R vH C ) i^i D ) C_ ( A vH B ) )
28	27	ad2antll 765	. . . . . . . . 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 ) )
29	28	ad2antlr 763	. . . . . . . 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 ) )
30		sstr 3611	. . . . . . . . . . . . . 14 |- ( ( R C_ D /\ D C_ ( A vH B ) ) -> R C_ ( A vH B ) )
31	30	ancoms 469	. . . . . . . . . . . . 13 |- ( ( D C_ ( A vH B ) /\ R C_ D ) -> R C_ ( A vH B ) )
32	31	ad2ant2l 782	. . . . . . . . . . . 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 ) )
33	32	adantll 750	. . . . . . . . . . 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 ) )
34	33	adantll 750	. . . . . . . . . 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 ) )
35		ssinss1 3841	. . . . . . . . . . . 12 |- ( C C_ ( A vH B ) -> ( C i^i D ) C_ ( A vH B ) )
36	35	ad2antrl 764	. . . . . . . . . . 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 ) )
37	36	ad2antlr 763	. . . . . . . . . 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 ) )
38	34, 37	jca 554	. . . . . . . . 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 ) ) )
39		mdslmd.1	. . . . . . . . . . 11 |- A e. CH
40		mdslmd.2	. . . . . . . . . . 11 |- B e. CH
41	39, 40	chjcli 28316	. . . . . . . . . 10 |- ( A vH B ) e. CH
42	6, 17, 41	chlubi 28330	. . . . . . . . 9 |- ( ( R C_ ( A vH B ) /\ ( C i^i D ) C_ ( A vH B ) ) <-> ( R vH ( C i^i D ) ) C_ ( A vH B ) )
43	38, 42	sylib 208	. . . . . . . 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 ) )
44	29, 43	jca 554	. . . . . . 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 ) ) )
45	6, 5	chjcli 28316	. . . . . . . . 9 |- ( R vH C ) e. CH
46	45, 16	chincli 28319	. . . . . . . 8 |- ( ( R vH C ) i^i D ) e. CH
47	6, 17	chjcli 28316	. . . . . . . 8 |- ( R vH ( C i^i D ) ) e. CH
48	46, 47, 41	chlubi 28330	. . . . . . 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 ) )
49	44, 48	sylib 208	. . . . . 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 ) )
50	39, 40, 46, 47	mdslle1i 29176	. . . . . 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 ) ) )
51	4, 24, 49, 50	syl3anc 1326	. . . . 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 ) ) )
52		inindir 3831	. . . . . . 7 |- ( ( ( R vH C ) i^i D ) i^i B ) = ( ( ( R vH C ) i^i B ) i^i ( D i^i B ) )
53		sstr 3611	. . . . . . . . . . . . . 14 |- ( ( A C_ ( C i^i D ) /\ ( C i^i D ) C_ R ) -> A C_ R )
54	15, 53	sylanb 489	. . . . . . . . . . . . 13 |- ( ( ( A C_ C /\ A C_ D ) /\ ( C i^i D ) C_ R ) -> A C_ R )
55	54	ad2ant2r 783	. . . . . . . . . . . 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 )
56		simplll 798	. . . . . . . . . . . 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_ C )
57	55, 56	ssind 3837	. . . . . . . . . . 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 ) )
58		simplrl 800	. . . . . . . . . . . . 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 ) )
59	33, 58	jca 554	. . . . . . . . . . . 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 ) ) )
60	6, 5, 41	chlubi 28330	. . . . . . . . . . . 12 |- ( ( R C_ ( A vH B ) /\ C C_ ( A vH B ) ) <-> ( R vH C ) C_ ( A vH B ) )
61	59, 60	sylib 208	. . . . . . . . . . 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 ) )
62	57, 61	jca 554	. . . . . . . . . 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 ) ) )
63	39, 40, 6, 5	mdslj1i 29178	. . . . . . . . . 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 ) ) )
64	62, 63	sylan2 491	. . . . . . . . 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 ) ) )
65	64	anassrs 680	. . . . . . . 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 ) ) )
66	65	ineq1d 3813	. . . . . . 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 ) ) )
67	52, 66	syl5req 2669	. . . . . 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 ) )
68	15	biimpi 206	. . . . . . . . . . . . 13 |- ( ( A C_ C /\ A C_ D ) -> A C_ ( C i^i D ) )
69	68	adantr 481	. . . . . . . . . . . 12 |- ( ( ( A C_ C /\ A C_ D ) /\ ( C i^i D ) C_ R ) -> A C_ ( C i^i D ) )
70	54, 69	ssind 3837	. . . . . . . . . . 11 |- ( ( ( A C_ C /\ A C_ D ) /\ ( C i^i D ) C_ R ) -> A C_ ( R i^i ( C i^i D ) ) )
71	31	adantll 750	. . . . . . . . . . . . 13 |- ( ( ( C C_ ( A vH B ) /\ D C_ ( A vH B ) ) /\ R C_ D ) -> R C_ ( A vH B ) )
72	35	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( C C_ ( A vH B ) /\ D C_ ( A vH B ) ) /\ R C_ D ) -> ( C i^i D ) C_ ( A vH B ) )
73	71, 72	jca 554	. . . . . . . . . . . 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 ) ) )
74	73, 42	sylib 208	. . . . . . . . . . 11 |- ( ( ( C C_ ( A vH B ) /\ D C_ ( A vH B ) ) /\ R C_ D ) -> ( R vH ( C i^i D ) ) C_ ( A vH B ) )
75	70, 74	anim12i 590	. . . . . . . . . 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 ) ) )
76	75	an4s 869	. . . . . . . . 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 ) ) )
77	39, 40, 6, 17	mdslj1i 29178	. . . . . . . . 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 ) ) )
78	76, 77	sylan2 491	. . . . . . . 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 ) ) )
79	78	anassrs 680	. . . . . . 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 ) ) )
80		inindir 3831	. . . . . . . . 9 |- ( ( C i^i D ) i^i B ) = ( ( C i^i B ) i^i ( D i^i B ) )
81	80	a1i 11	. . . . . . . 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 ) ) )
82	81	oveq2d 6666	. . . . . . 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 ) ) ) )
83	79, 82	eqtr2d 2657	. . . . . 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 ) )
84	67, 83	sseq12d 3634	. . . . 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 ) ) )
85	51, 84	bitr4d 271	. . . 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 ) ) ) ) )
86	85	exbiri 652	. . 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 ) ) ) ) )
87	86	a2d 29	. 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 ) ) ) ) )
88	3, 87	syl5 34	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 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CHcch 27786   vH chj 27790   MH cmd 27823   MH* cdmd 27824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-lm 21033  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cfil 23053  df-cau 23054  df-cmet 23055  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-dip 27556  df-ssp 27577  df-ph 27668  df-cbn 27719  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830  df-sh 28064  df-ch 28078  df-oc 28109  df-ch0 28110  df-shs 28167  df-chj 28169  df-md 29139  df-dmd 29140

This theorem is referenced by:  mdslmd1lem4  29187