Theorem mulextsr1lem 6956

Description: Lemma for mulextsr1 6957. (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 6768	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
2	1	adantl 271	. . . . . 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 6727	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
4	3	adantl 271	. . . . . . 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 964	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> Z e. P. )
6		simp3r 967	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> V e. P. )
7		mulclpr 6762	. . . . . . . 8 |- ( ( Z e. P. /\ V e. P. ) -> ( Z .P. V ) e. P. )
8	5, 6, 7	syl2anc 403	. . . . . . 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 963	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> Y e. P. )
10		mulclpr 6762	. . . . . . . 8 |- ( ( Y e. P. /\ V e. P. ) -> ( Y .P. V ) e. P. )
11	9, 6, 10	syl2anc 403	. . . . . . 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 5674	. . . . . 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 962	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> X e. P. )
14		simp3l 966	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> U e. P. )
15		mulclpr 6762	. . . . . . . 8 |- ( ( X e. P. /\ U e. P. ) -> ( X .P. U ) e. P. )
16	13, 14, 15	syl2anc 403	. . . . . . 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 965	. . . . . . . 8 |- ( ( ( X e. P. /\ Y e. P. ) /\ ( Z e. P. /\ W e. P. ) /\ ( U e. P. /\ V e. P. ) ) -> W e. P. )
18		mulclpr 6762	. . . . . . . 8 |- ( ( W e. P. /\ U e. P. ) -> ( W .P. U ) e. P. )
19	17, 14, 18	syl2anc 403	. . . . . . 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 5674	. . . . . 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 5677	. . . . 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 6769	. . . . . . 7 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
23	22	adantl 271	. . . . . 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 5706	. . . . 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 6778	. . . . . . . 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 271	. . . . . . 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 6770	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
28	27	adantl 271	. . . . . . 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 5697	. . . . . 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 5697	. . . . . 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 5550	. . . . 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 2123	. . . 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 6762	. . . . . . 7 |- ( ( X e. P. /\ V e. P. ) -> ( X .P. V ) e. P. )
34	13, 6, 33	syl2anc 403	. . . . . 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 6762	. . . . . . 7 |- ( ( Y e. P. /\ U e. P. ) -> ( Y .P. U ) e. P. )
36	9, 14, 35	syl2anc 403	. . . . . 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 6762	. . . . . . 7 |- ( ( Z e. P. /\ U e. P. ) -> ( Z .P. U ) e. P. )
38	5, 14, 37	syl2anc 403	. . . . . 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 6762	. . . . . . 7 |- ( ( W e. P. /\ V e. P. ) -> ( W .P. V ) e. P. )
40	17, 6, 39	syl2anc 403	. . . . . 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 5706	. . . . 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 5697	. . . . . 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 5697	. . . . . 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 5550	. . . . 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 2116	. . . 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 3798	. . 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 2155	. . . . 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 2155	. . . . 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 6727	. . . . . . 7 |- ( ( Z e. P. /\ Y e. P. ) -> ( Z +P. Y ) e. P. )
50	5, 9, 49	syl2anc 403	. . . . . 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 6762	. . . . . 6 |- ( ( ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( Z +P. Y ) .P. U ) e. P. )
52	50, 14, 51	syl2anc 403	. . . . 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 6727	. . . . . . 7 |- ( ( X e. P. /\ W e. P. ) -> ( X +P. W ) e. P. )
54	13, 17, 53	syl2anc 403	. . . . . 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 6762	. . . . . 6 |- ( ( ( X +P. W ) e. P. /\ V e. P. ) -> ( ( X +P. W ) .P. V ) e. P. )
56	54, 6, 55	syl2anc 403	. . . . 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 6811	. . . . 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 1170	. . . 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 6770	. . . . . . . . 9 |- ( ( ( X +P. W ) e. P. /\ U e. P. ) -> ( ( X +P. W ) .P. U ) = ( U .P. ( X +P. W ) ) )
60	59	3adant2 957	. . . . . . . 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 6770	. . . . . . . . 9 |- ( ( ( Z +P. Y ) e. P. /\ U e. P. ) -> ( ( Z +P. Y ) .P. U ) = ( U .P. ( Z +P. Y ) ) )
62	61	3adant1 956	. . . . . . . 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 3798	. . . . . . 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 6832	. . . . . . 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 148	. . . . . 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 1169	. . . . 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 6770	. . . . . . . 8 |- ( ( ( Z +P. Y ) e. P. /\ V e. P. ) -> ( ( Z +P. Y ) .P. V ) = ( V .P. ( Z +P. Y ) ) )
68	50, 6, 67	syl2anc 403	. . . . . . 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 6770	. . . . . . . 8 |- ( ( ( X +P. W ) e. P. /\ V e. P. ) -> ( ( X +P. W ) .P. V ) = ( V .P. ( X +P. W ) ) )
70	54, 6, 69	syl2anc 403	. . . . . . 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 3798	. . . . . 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 6832	. . . . . . 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 1169	. . . . . 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 148	. . . . 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 732	. . . 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 44	. . 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 148	. 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 6768	. . . . 5 |- ( ( Z e. P. /\ Y e. P. ) -> ( Z +P. Y ) = ( Y +P. Z ) )
79	5, 9, 78	syl2anc 403	. . . 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 3797	. . 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 6768	. . . . 5 |- ( ( X e. P. /\ W e. P. ) -> ( X +P. W ) = ( W +P. X ) )
82	13, 17, 81	syl2anc 403	. . . 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 3797	. . 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 739	. 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 147	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 102   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   P.cnp 6481   +P. cpp 6483   .P. cmp 6484   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-imp 6659  df-iltp 6660

This theorem is referenced by:  mulextsr1  6957