Theorem mullocprlem 7927

Description: Calculations for mullocpr 7928. (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 7740	. . . . . . . . 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 7741	. . . . . . . . 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 6263	. . . . . . 7 |- ( ph -> ( E .Q ( D .Q U ) ) = ( U .Q ( D .Q E ) ) )
12	1, 11	breqtrd 4151	. . . . . 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 7733	. . . . . . . 8 |- ( ( D e. Q. /\ E e. Q. ) -> ( D .Q E ) e. Q. )
16	5, 3, 15	syl2anc 415	. . . . . . 7 |- ( ph -> ( D .Q E ) e. Q. )
17		ltmnqg 7758	. . . . . . 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 1278	. . . . . 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 7925	. . . . 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 1277	. . . 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 745	. 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 7740	. . . . . . 7 |- ( ( T e. Q. /\ U e. Q. ) -> ( T .Q U ) = ( U .Q T ) )
36	34, 6, 35	syl2anc 415	. . . . . 6 |- ( ph -> ( T .Q U ) = ( U .Q T ) )
37		mullocprlem.tdudr	. . . . . . 7 |- ( ph -> ( T .Q ( D .Q U ) ) <Q ( D .Q R ) )
38		mulclnq 7733	. . . . . . . . . 10 |- ( ( T e. Q. /\ U e. Q. ) -> ( T .Q U ) e. Q. )
39	34, 6, 38	syl2anc 415	. . . . . . . . 9 |- ( ph -> ( T .Q U ) e. Q. )
40	13	simprd 114	. . . . . . . . 9 |- ( ph -> R e. Q. )
41		ltmnqg 7758	. . . . . . . . 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 1278	. . . . . . . 8 |- ( ph -> ( ( T .Q U ) <Q R <-> ( D .Q ( T .Q U ) ) <Q ( D .Q R ) ) )
43	34, 5, 6, 8, 10	caov12d 6261	. . . . . . . . 9 |- ( ph -> ( T .Q ( D .Q U ) ) = ( D .Q ( T .Q U ) ) )
44	43	breq1d 4135	. . . . . . . 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 4149	. . . . 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 7926	. . . . 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 1277	. . . 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 746	. 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 7733	. . . . . . 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 7758	. . . . . 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 1278	. . . . 5 |- ( ph -> ( E <Q T <-> ( ( D .Q U ) .Q E ) <Q ( ( D .Q U ) .Q T ) ) )
63		mulcomnqg 7740	. . . . . . 7 |- ( ( ( D .Q U ) e. Q. /\ E e. Q. ) -> ( ( D .Q U ) .Q E ) = ( E .Q ( D .Q U ) ) )
64	60, 3, 63	syl2anc 415	. . . . . 6 |- ( ph -> ( ( D .Q U ) .Q E ) = ( E .Q ( D .Q U ) ) )
65		mulcomnqg 7740	. . . . . . 7 |- ( ( ( D .Q U ) e. Q. /\ T e. Q. ) -> ( ( D .Q U ) .Q T ) = ( T .Q ( D .Q U ) ) )
66	60, 34, 65	syl2anc 415	. . . . . 6 |- ( ph -> ( ( D .Q U ) .Q T ) = ( T .Q ( D .Q U ) ) )
67	64, 66	breq12d 4138	. . . . 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 7832	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
71		prloc 7848	. . . 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 415	. 2 |- ( ph -> ( E e. ( 1st ` B ) \/ T e. ( 2nd ` B ) ) )
74	33, 57, 73	mpjaodan 810	1 |- ( ph -> ( Q e. ( 1st ` ( A .P. B ) ) \/ R e. ( 2nd ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   .Q cmq 7640   <Q cltq 7642   P.cnp 7648   .P. cmp 7651

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-lti 7664  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823  df-imp 7826

This theorem is referenced by:  mullocpr  7928