Theorem mulextsr1lem 7990

Description: Lemma for mulextsr1 7991. (Contributed by Jim Kingdon, 17-Feb-2020.)

Assertion

Ref	Expression
mulextsr1lem	|- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) <P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) -> ( ( X +P. W ) <P ( Y +P. Z ) \/ ( Z +P. Y ) <P ( W +P. X ) ) ) )

Proof of Theorem mulextsr1lem

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcomprg 7788	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
2	1	adantl 277	. . . . . 6 |- ( ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
3		addclpr 7747	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
4	3	adantl 277	. . . . . . 7 |- ( ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) e. P. )
5		simp2l 1047	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> Z e. P. )
6		simp3r 1050	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> V e. P. )
7		mulclpr 7782	. . . . . . . 8 |- ( ( Z e. P. /\ V e. P. ) -> ( Z .P. V ) e. P. )
8	5, 6, 7	syl2anc 411	. . . . . . 7 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( Z .P. V ) e. P. )
9		simp1r 1046	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> Y e. P. )
10		mulclpr 7782	. . . . . . . 8 |- ( ( Y e. P. /\ V e. P. ) -> ( Y .P. V ) e. P. )
11	9, 6, 10	syl2anc 411	. . . . . . 7 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( Y .P. V ) e. P. )
12	4, 8, 11	caovcld 6171	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z .P. V ) +P. ( Y .P. V ) ) e. P. )
13		simp1l 1045	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> X e. P. )
14		simp3l 1049	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> U e. P. )
15		mulclpr 7782	. . . . . . . 8 |- ( ( X e. P. /\ U e. P. ) -> ( X .P. U ) e. P. )
16	13, 14, 15	syl2anc 411	. . . . . . 7 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( X .P. U ) e. P. )
17		simp2r 1048	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> W e. P. )
18		mulclpr 7782	. . . . . . . 8 |- ( ( W e. P. /\ U e. P. ) -> ( W .P. U ) e. P. )
19	17, 14, 18	syl2anc 411	. . . . . . 7 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( W .P. U ) e. P. )
20	4, 16, 19	caovcld 6171	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X .P. U ) +P. ( W .P. U ) ) e. P. )
21	2, 12, 20	caovcomd 6174	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( Z .P. V ) +P. ( Y .P. V ) ) +P. ( ( X .P. U ) +P. ( W .P. U ) ) ) = ( ( ( X .P. U ) +P. ( W .P. U ) ) +P. ( ( Z .P. V ) +P. ( Y .P. V ) ) ) )
22		addassprg 7789	. . . . . . 7 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
23	22	adantl 277	. . . . . 6 |- ( ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
24	16, 11, 8, 2, 23, 19, 4	caov411d 6203	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) = ( ( ( Z .P. V ) +P. ( Y .P. V ) ) +P. ( ( X .P. U ) +P. ( W .P. U ) ) ) )
25		distrprg 7798	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f .P. ( g +P. h ) ) = ( ( f .P. g ) +P. ( f .P. h ) ) )
26	25	adantl 277	. . . . . . 7 |- ( ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f .P. ( g +P. h ) ) = ( ( f .P. g ) +P. ( f .P. h ) ) )
27		mulcomprg 7790	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
28	27	adantl 277	. . . . . . 7 |- ( ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f .P. g ) = ( g .P. f ) )
29	26, 13, 17, 14, 4, 28	caovdir2d 6194	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X +P. W ) .P. U ) = ( ( X .P. U ) +P. ( W .P. U ) ) )
30	26, 5, 9, 6, 4, 28	caovdir2d 6194	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z +P. Y ) .P. V ) = ( ( Z .P. V ) +P. ( Y .P. V ) ) )
31	29, 30	oveq12d 6031	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X +P. W ) .P. U ) +P. ( ( Z +P. Y ) .P. V ) ) = ( ( ( X .P. U ) +P. ( W .P. U ) ) +P. ( ( Z .P. V ) +P. ( Y .P. V ) ) ) )
32	21, 24, 31	3eqtr4d 2272	. . . 4 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) = ( ( ( X +P. W ) .P. U ) +P. ( ( Z +P. Y ) .P. V ) ) )
33		mulclpr 7782	. . . . . . 7 |- ( ( X e. P. /\ V e. P. ) -> ( X .P. V ) e. P. )
34	13, 6, 33	syl2anc 411	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( X .P. V ) e. P. )
35		mulclpr 7782	. . . . . . 7 |- ( ( Y e. P. /\ U e. P. ) -> ( Y .P. U ) e. P. )
36	9, 14, 35	syl2anc 411	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( Y .P. U ) e. P. )
37		mulclpr 7782	. . . . . . 7 |- ( ( Z e. P. /\ U e. P. ) -> ( Z .P. U ) e. P. )
38	5, 14, 37	syl2anc 411	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( Z .P. U ) e. P. )
39		mulclpr 7782	. . . . . . 7 |- ( ( W e. P. /\ V e. P. ) -> ( W .P. V ) e. P. )
40	17, 6, 39	syl2anc 411	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( W .P. V ) e. P. )
41	34, 36, 38, 2, 23, 40, 4	caov411d 6203	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) = ( ( ( Z .P. U ) +P. ( Y .P. U ) ) +P. ( ( X .P. V ) +P. ( W .P. V ) ) ) )
42	26, 5, 9, 14, 4, 28	caovdir2d 6194	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z +P. Y ) .P. U ) = ( ( Z .P. U ) +P. ( Y .P. U ) ) )
43	26, 13, 17, 6, 4, 28	caovdir2d 6194	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X +P. W ) .P. V ) = ( ( X .P. V ) +P. ( W .P. V ) ) )
44	42, 43	oveq12d 6031	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( Z +P. Y ) .P. U ) +P. ( ( X +P. W ) .P. V ) ) = ( ( ( Z .P. U ) +P. ( Y .P. U ) ) +P. ( ( X .P. V ) +P. ( W .P. V ) ) ) )
45	41, 44	eqtr4d 2265	. . . 4 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) = ( ( ( Z +P. Y ) .P. U ) +P. ( ( X +P. W ) .P. V ) ) )
46	32, 45	breq12d 4099	. . 3 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) <P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) <-> ( ( ( X +P. W ) .P. U ) +P. ( ( Z +P. Y ) .P. V ) ) <P ( ( ( Z +P. Y ) .P. U ) +P. ( ( X +P. W ) .P. V ) ) ) )
47	29, 20	eqeltrd 2306	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X +P. W ) .P. U ) e. P. )
48	30, 12	eqeltrd 2306	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z +P. Y ) .P. V ) e. P. )
49		addclpr 7747	. . . . . . 7 |- ( ( Z e. P. /\ Y e. P. ) -> ( Z +P. Y ) e. P. )
50	5, 9, 49	syl2anc 411	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( Z +P. Y ) e. P. )
51		mulclpr 7782	. . . . . 6 |- ( ( ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( Z +P. Y ) .P. U ) e. P. )
52	50, 14, 51	syl2anc 411	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z +P. Y ) .P. U ) e. P. )
53		addclpr 7747	. . . . . . 7 |- ( ( X e. P. /\ W e. P. ) -> ( X +P. W ) e. P. )
54	13, 17, 53	syl2anc 411	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( X +P. W ) e. P. )
55		mulclpr 7782	. . . . . 6 |- ( ( ( X +P. W ) e. P. /\ V e. P. ) -> ( ( X +P. W ) .P. V ) e. P. )
56	54, 6, 55	syl2anc 411	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X +P. W ) .P. V ) e. P. )
57		addextpr 7831	. . . . 5 |- ( ( ( ( ( X +P. W ) .P. U ) e. P. /\ ( ( Z +P. Y ) .P. V ) e. P. ) /\ ( ( ( Z +P. Y ) .P. U ) e. P. /\ ( ( X +P. W ) .P. V ) e. P. ) ) -> ( ( ( ( X +P. W ) .P. U ) +P. ( ( Z +P. Y ) .P. V ) ) <P ( ( ( Z +P. Y ) .P. U ) +P. ( ( X +P. W ) .P. V ) ) -> ( ( ( X +P. W ) .P. U ) <P ( ( Z +P. Y ) .P. U ) \/ ( ( Z +P. Y ) .P. V ) <P ( ( X +P. W ) .P. V ) ) ) )
58	47, 48, 52, 56, 57	syl22anc 1272	. . . 4 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X +P. W ) .P. U ) +P. ( ( Z +P. Y ) .P. V ) ) <P ( ( ( Z +P. Y ) .P. U ) +P. ( ( X +P. W ) .P. V ) ) -> ( ( ( X +P. W ) .P. U ) <P ( ( Z +P. Y ) .P. U ) \/ ( ( Z +P. Y ) .P. V ) <P ( ( X +P. W ) .P. V ) ) ) )
59		mulcomprg 7790	. . . . . . . . 9 |- ( ( ( X +P. W ) e. P. /\ U e. P. ) -> ( ( X +P. W ) .P. U ) = ( U .P. ( X +P. W ) ) )
60	59	3adant2 1040	. . . . . . . 8 |- ( ( ( X +P. W ) e. P. /\ ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( X +P. W ) .P. U ) = ( U .P. ( X +P. W ) ) )
61		mulcomprg 7790	. . . . . . . . 9 |- ( ( ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( Z +P. Y ) .P. U ) = ( U .P. ( Z +P. Y ) ) )
62	61	3adant1 1039	. . . . . . . 8 |- ( ( ( X +P. W ) e. P. /\ ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( Z +P. Y ) .P. U ) = ( U .P. ( Z +P. Y ) ) )
63	60, 62	breq12d 4099	. . . . . . 7 |- ( ( ( X +P. W ) e. P. /\ ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( ( X +P. W ) .P. U ) <P ( ( Z +P. Y ) .P. U ) <-> ( U .P. ( X +P. W ) ) <P ( U .P. ( Z +P. Y ) ) ) )
64		ltmprr 7852	. . . . . . 7 |- ( ( ( X +P. W ) e. P. /\ ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( U .P. ( X +P. W ) ) <P ( U .P. ( Z +P. Y ) ) -> ( X +P. W ) <P ( Z +P. Y ) ) )
65	63, 64	sylbid 150	. . . . . 6 |- ( ( ( X +P. W ) e. P. /\ ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( ( X +P. W ) .P. U ) <P ( ( Z +P. Y ) .P. U ) -> ( X +P. W ) <P ( Z +P. Y ) ) )
66	54, 50, 14, 65	syl3anc 1271	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X +P. W ) .P. U ) <P ( ( Z +P. Y ) .P. U ) -> ( X +P. W ) <P ( Z +P. Y ) ) )
67		mulcomprg 7790	. . . . . . . 8 |- ( ( ( Z +P. Y ) e. P. /\ V e. P. ) -> ( ( Z +P. Y ) .P. V ) = ( V .P. ( Z +P. Y ) ) )
68	50, 6, 67	syl2anc 411	. . . . . . 7 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z +P. Y ) .P. V ) = ( V .P. ( Z +P. Y ) ) )
69		mulcomprg 7790	. . . . . . . 8 |- ( ( ( X +P. W ) e. P. /\ V e. P. ) -> ( ( X +P. W ) .P. V ) = ( V .P. ( X +P. W ) ) )
70	54, 6, 69	syl2anc 411	. . . . . . 7 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X +P. W ) .P. V ) = ( V .P. ( X +P. W ) ) )
71	68, 70	breq12d 4099	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( Z +P. Y ) .P. V ) <P ( ( X +P. W ) .P. V ) <-> ( V .P. ( Z +P. Y ) ) <P ( V .P. ( X +P. W ) ) ) )
72		ltmprr 7852	. . . . . . 7 |- ( ( ( Z +P. Y ) e. P. /\ ( X +P. W ) e. P. /\ V e. P. ) -> ( ( V .P. ( Z +P. Y ) ) <P ( V .P. ( X +P. W ) ) -> ( Z +P. Y ) <P ( X +P. W ) ) )
73	50, 54, 6, 72	syl3anc 1271	. . . . . 6 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( V .P. ( Z +P. Y ) ) <P ( V .P. ( X +P. W ) ) -> ( Z +P. Y ) <P ( X +P. W ) ) )
74	71, 73	sylbid 150	. . . . 5 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( Z +P. Y ) .P. V ) <P ( ( X +P. W ) .P. V ) -> ( Z +P. Y ) <P ( X +P. W ) ) )
75	66, 74	orim12d 791	. . . 4 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X +P. W ) .P. U ) <P ( ( Z +P. Y ) .P. U ) \/ ( ( Z +P. Y ) .P. V ) <P ( ( X +P. W ) .P. V ) ) -> ( ( X +P. W ) <P ( Z +P. Y ) \/ ( Z +P. Y ) <P ( X +P. W ) ) ) )
76	58, 75	syld 45	. . 3 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X +P. W ) .P. U ) +P. ( ( Z +P. Y ) .P. V ) ) <P ( ( ( Z +P. Y ) .P. U ) +P. ( ( X +P. W ) .P. V ) ) -> ( ( X +P. W ) <P ( Z +P. Y ) \/ ( Z +P. Y ) <P ( X +P. W ) ) ) )
77	46, 76	sylbid 150	. 2 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) <P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) -> ( ( X +P. W ) <P ( Z +P. Y ) \/ ( Z +P. Y ) <P ( X +P. W ) ) ) )
78		addcomprg 7788	. . . . 5 |- ( ( Z e. P. /\ Y e. P. ) -> ( Z +P. Y ) = ( Y +P. Z ) )
79	5, 9, 78	syl2anc 411	. . . 4 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( Z +P. Y ) = ( Y +P. Z ) )
80	79	breq2d 4098	. . 3 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( X +P. W ) <P ( Z +P. Y ) <-> ( X +P. W ) <P ( Y +P. Z ) ) )
81		addcomprg 7788	. . . . 5 |- ( ( X e. P. /\ W e. P. ) -> ( X +P. W ) = ( W +P. X ) )
82	13, 17, 81	syl2anc 411	. . . 4 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( X +P. W ) = ( W +P. X ) )
83	82	breq2d 4098	. . 3 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( Z +P. Y ) <P ( X +P. W ) <-> ( Z +P. Y ) <P ( W +P. X ) ) )
84	80, 83	orbi12d 798	. 2 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( X +P. W ) <P ( Z +P. Y ) \/ ( Z +P. Y ) <P ( X +P. W ) ) <-> ( ( X +P. W ) <P ( Y +P. Z ) \/ ( Z +P. Y ) <P ( W +P. X ) ) ) )
85	77, 84	sylibd 149	1 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> ( ( ( ( X .P. U ) +P. ( Y .P. V ) ) +P. ( ( Z .P. V ) +P. ( W .P. U ) ) ) <P ( ( ( X .P. V ) +P. ( Y .P. U ) ) +P. ( ( Z .P. U ) +P. ( W .P. V ) ) ) -> ( ( X +P. W ) <P ( Y +P. Z ) \/ ( Z +P. Y ) <P ( W +P. X ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   P.cnp 7501   +P. cpp 7503   .P. cmp 7504   <P cltp 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-imp 7679  df-iltp 7680

This theorem is referenced by:  mulextsr1  7991