Theorem mullocprlem 7768

Description: Calculations for mullocpr 7769. (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 7581	. . . . . . . . 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 7582	. . . . . . . . 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 6195	. . . . . . 7 |- ( ph -> ( E .Q ( D .Q U ) ) = ( U .Q ( D .Q E ) ) )
12	1, 11	breqtrd 4109	. . . . . 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 7574	. . . . . . . 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 7599	. . . . . . 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 1271	. . . . . 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 7766	. . . . 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 1270	. . . 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 738	. 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 7581	. . . . . . 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 7574	. . . . . . . . . 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 7599	. . . . . . . . 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 1271	. . . . . . . 8 |- ( ph -> ( ( T .Q U ) <Q R <-> ( D .Q ( T .Q U ) ) <Q ( D .Q R ) ) )
43	34, 5, 6, 8, 10	caov12d 6193	. . . . . . . . 9 |- ( ph -> ( T .Q ( D .Q U ) ) = ( D .Q ( T .Q U ) ) )
44	43	breq1d 4093	. . . . . . . 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 4107	. . . . 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 7767	. . . . 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 1270	. . . 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 739	. 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 7574	. . . . . . 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 7599	. . . . . 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 1271	. . . . 5 |- ( ph -> ( E <Q T <-> ( ( D .Q U ) .Q E ) <Q ( ( D .Q U ) .Q T ) ) )
63		mulcomnqg 7581	. . . . . . 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 7581	. . . . . . 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 4096	. . . . 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 7673	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
71		prloc 7689	. . . 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 803	1 |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   1stc1st 6290   2ndc2nd 6291   Q.cnq 7478   .Q cmq 7481   <Q cltq 7483   P.cnp 7489   .P. cmp 7492

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-mi 7504  df-lti 7505  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-inp 7664  df-imp 7667

This theorem is referenced by:  mullocpr  7769