Theorem mullocprlem 7682

Description: Calculations for mullocpr 7683. (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 7495	. . . . . . . . 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 7496	. . . . . . . . 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 6129	. . . . . . 7 |- ( ph -> ( E .Q ( D .Q U ) ) = ( U .Q ( D .Q E ) ) )
12	1, 11	breqtrd 4069	. . . . . 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 7488	. . . . . . . 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 7513	. . . . . . 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 1249	. . . . . 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 7680	. . . . 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 1248	. . . 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 734	. 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 7495	. . . . . . 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 7488	. . . . . . . . . 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 7513	. . . . . . . . 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 1249	. . . . . . . 8 |- ( ph -> ( ( T .Q U ) <Q R <-> ( D .Q ( T .Q U ) ) <Q ( D .Q R ) ) )
43	34, 5, 6, 8, 10	caov12d 6127	. . . . . . . . 9 |- ( ph -> ( T .Q ( D .Q U ) ) = ( D .Q ( T .Q U ) ) )
44	43	breq1d 4053	. . . . . . . 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 4067	. . . . 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 7681	. . . . 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 1248	. . . 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 735	. 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 7488	. . . . . . 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 7513	. . . . . 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 1249	. . . . 5 |- ( ph -> ( E <Q T <-> ( ( D .Q U ) .Q E ) <Q ( ( D .Q U ) .Q T ) ) )
63		mulcomnqg 7495	. . . . . . 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 7495	. . . . . . 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 4056	. . . . 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 7587	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
71		prloc 7603	. . . 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 799	1 |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1372   e. wcel 2175   <.cop 3635   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   1stc1st 6223   2ndc2nd 6224   Q.cnq 7392   .Q cmq 7395   <Q cltq 7397   P.cnp 7403   .P. cmp 7406

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-mi 7418  df-lti 7419  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-inp 7578  df-imp 7581

This theorem is referenced by:  mullocpr  7683