Theorem mullocprlem 7901

Description: Calculations for mullocpr 7902. (Contributed by Jim Kingdon, 10-Dec-2019.)

Hypotheses

Ref	Expression
mullocprlem.ab	|- ( ph -> ( A e. P. /\ B e. P. ) )
mullocprlem.uqedu	|- ( ph -> ( U .Q Q ) <Q ( E .Q ( D .Q U ) ) )
mullocprlem.edutdu	|- ( ph -> ( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) )
mullocprlem.tdudr	|- ( ph -> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) )
mullocprlem.qr	|- ( ph -> ( Q e. Q. /\ R e. Q. ) )
mullocprlem.duq	|- ( ph -> ( D e. Q. /\ U e. Q. ) )
mullocprlem.du	|- ( ph -> ( D e. ( 1st ` A ) /\ U e. ( 2nd ` A ) ) )
mullocprlem.et	|- ( ph -> ( E e. Q. /\ T e. Q. ) )

Assertion

Ref	Expression
mullocprlem	|- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )

Proof of Theorem mullocprlem

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mullocprlem.uqedu	. . . . . . 7 |- ( ph -> ( U .Q Q ) <Q ( E .Q ( D .Q U ) ) )
2		mullocprlem.et	. . . . . . . . 9 |- ( ph -> ( E e. Q. /\ T e. Q. ) )
3	2	simpld 112	. . . . . . . 8 |- ( ph -> E e. Q. )
4		mullocprlem.duq	. . . . . . . . 9 |- ( ph -> ( D e. Q. /\ U e. Q. ) )
5	4	simpld 112	. . . . . . . 8 |- ( ph -> D e. Q. )
6	4	simprd 114	. . . . . . . 8 |- ( ph -> U e. Q. )
7		mulcomnqg 7714	. . . . . . . . 9 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) = ( y .Q x ) )
8	7	adantl 277	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ y e. Q. ) ) -> ( x .Q y ) = ( y .Q x ) )
9		mulassnqg 7715	. . . . . . . . 9 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( x .Q y ) .Q z ) = ( x .Q ( y .Q z ) ) )
10	9	adantl 277	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ y e. Q. /\ z e. Q. ) ) -> ( ( x .Q y ) .Q z ) = ( x .Q ( y .Q z ) ) )
11	3, 5, 6, 8, 10	caov13d 6246	. . . . . . 7 |- ( ph -> ( E .Q ( D .Q U ) ) = ( U .Q ( D .Q E ) ) )
12	1, 11	breqtrd 4140	. . . . . 6 |- ( ph -> ( U .Q Q ) <Q ( U .Q ( D .Q E ) ) )
13		mullocprlem.qr	. . . . . . . 8 |- ( ph -> ( Q e. Q. /\ R e. Q. ) )
14	13	simpld 112	. . . . . . 7 |- ( ph -> Q e. Q. )
15		mulclnq 7707	. . . . . . . 8 |- ( ( D e. Q. /\ E e. Q. ) -> ( D .Q E ) e. Q. )
16	5, 3, 15	syl2anc 411	. . . . . . 7 |- ( ph -> ( D .Q E ) e. Q. )
17		ltmnqg 7732	. . . . . . 7 |- ( ( Q e. Q. /\ ( D .Q E ) e. Q. /\ U e. Q. ) -> ( Q <Q ( D .Q E ) <-> ( U .Q Q ) <Q ( U .Q ( D .Q E ) ) ) )
18	14, 16, 6, 17	syl3anc 1274	. . . . . 6 |- ( ph -> ( Q <Q ( D .Q E ) <-> ( U .Q Q ) <Q ( U .Q ( D .Q E ) ) ) )
19	12, 18	mpbird 167	. . . . 5 |- ( ph -> Q <Q ( D .Q E ) )
20	19	adantr 276	. . . 4 |- ( ( ph /\ E e. ( 1st ` B ) ) -> Q <Q ( D .Q E ) )
21		mullocprlem.ab	. . . . . . . 8 |- ( ph -> ( A e. P. /\ B e. P. ) )
22	21	simpld 112	. . . . . . 7 |- ( ph -> A e. P. )
23		mullocprlem.du	. . . . . . . 8 |- ( ph -> ( D e. ( 1st ` A ) /\ U e. ( 2nd ` A ) ) )
24	23	simpld 112	. . . . . . 7 |- ( ph -> D e. ( 1st ` A ) )
25	22, 24	jca 306	. . . . . 6 |- ( ph -> ( A e. P. /\ D e. ( 1st ` A ) ) )
26	25	adantr 276	. . . . 5 |- ( ( ph /\ E e. ( 1st ` B ) ) -> ( A e. P. /\ D e. ( 1st ` A ) ) )
27	21	simprd 114	. . . . . 6 |- ( ph -> B e. P. )
28	27	anim1i 340	. . . . 5 |- ( ( ph /\ E e. ( 1st ` B ) ) -> ( B e. P. /\ E e. ( 1st ` B ) ) )
29	14	adantr 276	. . . . 5 |- ( ( ph /\ E e. ( 1st ` B ) ) -> Q e. Q. )
30		mulnqprl 7899	. . . . 5 |- ( ( ( ( A e. P. /\ D e. ( 1st ` A ) ) /\ ( B e. P. /\ E e. ( 1st ` B ) ) ) /\ Q e. Q. ) -> ( Q <Q ( D .Q E ) -> Q e. ( 1st ` ( A .P. B ) ) ) )
31	26, 28, 29, 30	syl21anc 1273	. . . 4 |- ( ( ph /\ E e. ( 1st ` B ) ) -> ( Q <Q ( D .Q E ) -> Q e. ( 1st ` ( A .P. B ) ) ) )
32	20, 31	mpd 13	. . 3 |- ( ( ph /\ E e. ( 1st ` B ) ) -> Q e. ( 1st ` ( A .P. B ) ) )
33	32	orcd 741	. 2 |- ( ( ph /\ E e. ( 1st ` B ) ) -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )
34	2	simprd 114	. . . . . . 7 |- ( ph -> T e. Q. )
35		mulcomnqg 7714	. . . . . . 7 |- ( ( T e. Q. /\ U e. Q. ) -> ( T .Q U ) = ( U .Q T ) )
36	34, 6, 35	syl2anc 411	. . . . . 6 |- ( ph -> ( T .Q U ) = ( U .Q T ) )
37		mullocprlem.tdudr	. . . . . . 7 |- ( ph -> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) )
38		mulclnq 7707	. . . . . . . . . 10 |- ( ( T e. Q. /\ U e. Q. ) -> ( T .Q U ) e. Q. )
39	34, 6, 38	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( T .Q U ) e. Q. )
40	13	simprd 114	. . . . . . . . 9 |- ( ph -> R e. Q. )
41		ltmnqg 7732	. . . . . . . . 9 |- ( ( ( T .Q U ) e. Q. /\ R e. Q. /\ D e. Q. ) -> ( ( T .Q U ) <Q R <-> ( D .Q ( T .Q U ) ) <Q ( D .Q R ) ) )
42	39, 40, 5, 41	syl3anc 1274	. . . . . . . 8 |- ( ph -> ( ( T .Q U ) <Q R <-> ( D .Q ( T .Q U ) ) <Q ( D .Q R ) ) )
43	34, 5, 6, 8, 10	caov12d 6244	. . . . . . . . 9 |- ( ph -> ( T .Q ( D .Q U ) ) = ( D .Q ( T .Q U ) ) )
44	43	breq1d 4124	. . . . . . . 8 |- ( ph -> ( ( T .Q ( D .Q U ) ) <Q ( D .Q R ) <-> ( D .Q ( T .Q U ) ) <Q ( D .Q R ) ) )
45	42, 44	bitr4d 191	. . . . . . 7 |- ( ph -> ( ( T .Q U ) <Q R <-> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) ) )
46	37, 45	mpbird 167	. . . . . 6 |- ( ph -> ( T .Q U ) <Q R )
47	36, 46	eqbrtrrd 4138	. . . . 5 |- ( ph -> ( U .Q T ) <Q R )
48	47	adantr 276	. . . 4 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> ( U .Q T ) <Q R )
49	23	simprd 114	. . . . . . 7 |- ( ph -> U e. ( 2nd ` A ) )
50	22, 49	jca 306	. . . . . 6 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
51	50	adantr 276	. . . . 5 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
52	27	anim1i 340	. . . . 5 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
53	40	adantr 276	. . . . 5 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> R e. Q. )
54		mulnqpru 7900	. . . . 5 |- ( ( ( ( A e. P. /\ U e. ( 2nd ` A ) ) /\ ( B e. P. /\ T e. ( 2nd ` B ) ) ) /\ R e. Q. ) -> ( ( U .Q T ) <Q R -> R e. ( 2nd ` ( A .P. B ) ) ) )
55	51, 52, 53, 54	syl21anc 1273	. . . 4 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> ( ( U .Q T ) <Q R -> R e. ( 2nd ` ( A .P. B ) ) ) )
56	48, 55	mpd 13	. . 3 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> R e. ( 2nd ` ( A .P. B ) ) )
57	56	olcd 742	. 2 |- ( ( ph /\ T e. ( 2nd ` B ) ) -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )
58		mullocprlem.edutdu	. . . 4 |- ( ph -> ( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) )
59		mulclnq 7707	. . . . . . 7 |- ( ( D e. Q. /\ U e. Q. ) -> ( D .Q U ) e. Q. )
60	4, 59	syl 14	. . . . . 6 |- ( ph -> ( D .Q U ) e. Q. )
61		ltmnqg 7732	. . . . . 6 |- ( ( E e. Q. /\ T e. Q. /\ ( D .Q U ) e. Q. ) -> ( E <Q T <-> ( ( D .Q U ) .Q E ) <Q ( ( D .Q U ) .Q T ) ) )
62	3, 34, 60, 61	syl3anc 1274	. . . . 5 |- ( ph -> ( E <Q T <-> ( ( D .Q U ) .Q E ) <Q ( ( D .Q U ) .Q T ) ) )
63		mulcomnqg 7714	. . . . . . 7 |- ( ( ( D .Q U ) e. Q. /\ E e. Q. ) -> ( ( D .Q U ) .Q E ) = ( E .Q ( D .Q U ) ) )
64	60, 3, 63	syl2anc 411	. . . . . 6 |- ( ph -> ( ( D .Q U ) .Q E ) = ( E .Q ( D .Q U ) ) )
65		mulcomnqg 7714	. . . . . . 7 |- ( ( ( D .Q U ) e. Q. /\ T e. Q. ) -> ( ( D .Q U ) .Q T ) = ( T .Q ( D .Q U ) ) )
66	60, 34, 65	syl2anc 411	. . . . . 6 |- ( ph -> ( ( D .Q U ) .Q T ) = ( T .Q ( D .Q U ) ) )
67	64, 66	breq12d 4127	. . . . 5 |- ( ph -> ( ( ( D .Q U ) .Q E ) <Q ( ( D .Q U ) .Q T ) <-> ( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) ) )
68	62, 67	bitrd 188	. . . 4 |- ( ph -> ( E <Q T <-> ( E .Q ( D .Q U ) ) <Q ( T .Q ( D .Q U ) ) ) )
69	58, 68	mpbird 167	. . 3 |- ( ph -> E <Q T )
70		prop 7806	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
71		prloc 7822	. . . 4 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E <Q T ) -> ( E e. ( 1st ` B ) \/ T e. ( 2nd ` B ) ) )
72	70, 71	sylan 283	. . 3 |- ( ( B e. P. /\ E <Q T ) -> ( E e. ( 1st ` B ) \/ T e. ( 2nd ` B ) ) )
73	27, 69, 72	syl2anc 411	. 2 |- ( ph -> ( E e. ( 1st ` B ) \/ T e. ( 2nd ` B ) ) )
74	33, 57, 73	mpjaodan 806	1 |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   <.cop 3697   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   .Q cmq 7614   <Q cltq 7616   P.cnp 7622   .P. cmp 7625

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-mi 7637  df-lti 7638  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-inp 7797  df-imp 7800

This theorem is referenced by:  mullocpr  7902