Theorem ltmnq 9794

Description: Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltmnq	|- ( C e. Q. -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )

Proof of Theorem ltmnq

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

Step	Hyp	Ref	Expression
1		mulnqf 9771	. . 3 |- .Q : ( Q. X. Q. ) --> Q.
2	1	fdmi 6052	. 2 |- dom .Q = ( Q. X. Q. )
3		ltrelnq 9748	. 2 |- <Q C_ ( Q. X. Q. )
4		0nnq 9746	. 2 |- -. (/) e. Q.
5		elpqn 9747	. . . . . . . . . 10 |- ( C e. Q. -> C e. ( N. X. N. ) )
6	5	3ad2ant3 1084	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
7		xp1st 7198	. . . . . . . . 9 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
8	6, 7	syl 17	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` C ) e. N. )
9		xp2nd 7199	. . . . . . . . 9 |- ( C e. ( N. X. N. ) -> ( 2nd ` C ) e. N. )
10	6, 9	syl 17	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` C ) e. N. )
11		mulclpi 9715	. . . . . . . 8 |- ( ( ( 1st ` C ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 1st ` C ) .N ( 2nd ` C ) ) e. N. )
12	8, 10, 11	syl2anc 693	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` C ) .N ( 2nd ` C ) ) e. N. )
13		ltmpi 9726	. . . . . . 7 |- ( ( ( 1st ` C ) .N ( 2nd ` C ) ) e. N. -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) )
14	12, 13	syl 17	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) )
15		fvex 6201	. . . . . . . 8 |- ( 1st ` C ) e. _V
16		fvex 6201	. . . . . . . 8 |- ( 2nd ` C ) e. _V
17		fvex 6201	. . . . . . . 8 |- ( 1st ` A ) e. _V
18		mulcompi 9718	. . . . . . . 8 |- ( x .N y ) = ( y .N x )
19		mulasspi 9719	. . . . . . . 8 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
20		fvex 6201	. . . . . . . 8 |- ( 2nd ` B ) e. _V
21	15, 16, 17, 18, 19, 20	caov4 6865	. . . . . . 7 |- ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 1st ` C ) .N ( 1st ` A ) ) .N ( ( 2nd ` C ) .N ( 2nd ` B ) ) )
22		fvex 6201	. . . . . . . 8 |- ( 1st ` B ) e. _V
23		fvex 6201	. . . . . . . 8 |- ( 2nd ` A ) e. _V
24	15, 16, 22, 18, 19, 23	caov4 6865	. . . . . . 7 |- ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) = ( ( ( 1st ` C ) .N ( 1st ` B ) ) .N ( ( 2nd ` C ) .N ( 2nd ` A ) ) )
25	21, 24	breq12i 4662	. . . . . 6 |- ( ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` C ) .N ( 2nd ` C ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) <-> ( ( ( 1st ` C ) .N ( 1st ` A ) ) .N ( ( 2nd ` C ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` C ) .N ( 1st ` B ) ) .N ( ( 2nd ` C ) .N ( 2nd ` A ) ) ) )
26	14, 25	syl6bb 276	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> ( ( ( 1st ` C ) .N ( 1st ` A ) ) .N ( ( 2nd ` C ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` C ) .N ( 1st ` B ) ) .N ( ( 2nd ` C ) .N ( 2nd ` A ) ) ) ) )
27		ordpipq 9764	. . . . 5 |- ( <. ( ( 1st ` C ) .N ( 1st ` A ) ) , ( ( 2nd ` C ) .N ( 2nd ` A ) ) >. <pQ <. ( ( 1st ` C ) .N ( 1st ` B ) ) , ( ( 2nd ` C ) .N ( 2nd ` B ) ) >. <-> ( ( ( 1st ` C ) .N ( 1st ` A ) ) .N ( ( 2nd ` C ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` C ) .N ( 1st ` B ) ) .N ( ( 2nd ` C ) .N ( 2nd ` A ) ) ) )
28	26, 27	syl6bbr 278	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> <. ( ( 1st ` C ) .N ( 1st ` A ) ) , ( ( 2nd ` C ) .N ( 2nd ` A ) ) >. <pQ <. ( ( 1st ` C ) .N ( 1st ` B ) ) , ( ( 2nd ` C ) .N ( 2nd ` B ) ) >. ) )
29		elpqn 9747	. . . . . . 7 |- ( A e. Q. -> A e. ( N. X. N. ) )
30	29	3ad2ant1 1082	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
31		mulpipq2 9761	. . . . . 6 |- ( ( C e. ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> ( C .pQ A ) = <. ( ( 1st ` C ) .N ( 1st ` A ) ) , ( ( 2nd ` C ) .N ( 2nd ` A ) ) >. )
32	6, 30, 31	syl2anc 693	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( C .pQ A ) = <. ( ( 1st ` C ) .N ( 1st ` A ) ) , ( ( 2nd ` C ) .N ( 2nd ` A ) ) >. )
33		elpqn 9747	. . . . . . 7 |- ( B e. Q. -> B e. ( N. X. N. ) )
34	33	3ad2ant2 1083	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
35		mulpipq2 9761	. . . . . 6 |- ( ( C e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( C .pQ B ) = <. ( ( 1st ` C ) .N ( 1st ` B ) ) , ( ( 2nd ` C ) .N ( 2nd ` B ) ) >. )
36	6, 34, 35	syl2anc 693	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( C .pQ B ) = <. ( ( 1st ` C ) .N ( 1st ` B ) ) , ( ( 2nd ` C ) .N ( 2nd ` B ) ) >. )
37	32, 36	breq12d 4666	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( C .pQ A ) <pQ ( C .pQ B ) <-> <. ( ( 1st ` C ) .N ( 1st ` A ) ) , ( ( 2nd ` C ) .N ( 2nd ` A ) ) >. <pQ <. ( ( 1st ` C ) .N ( 1st ` B ) ) , ( ( 2nd ` C ) .N ( 2nd ` B ) ) >. ) )
38	28, 37	bitr4d 271	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> ( C .pQ A ) <pQ ( C .pQ B ) ) )
39		ordpinq 9765	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
40	39	3adant3 1081	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
41		mulpqnq 9763	. . . . . . 7 |- ( ( C e. Q. /\ A e. Q. ) -> ( C .Q A ) = ( /Q ` ( C .pQ A ) ) )
42	41	ancoms 469	. . . . . 6 |- ( ( A e. Q. /\ C e. Q. ) -> ( C .Q A ) = ( /Q ` ( C .pQ A ) ) )
43	42	3adant2 1080	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( C .Q A ) = ( /Q ` ( C .pQ A ) ) )
44		mulpqnq 9763	. . . . . . 7 |- ( ( C e. Q. /\ B e. Q. ) -> ( C .Q B ) = ( /Q ` ( C .pQ B ) ) )
45	44	ancoms 469	. . . . . 6 |- ( ( B e. Q. /\ C e. Q. ) -> ( C .Q B ) = ( /Q ` ( C .pQ B ) ) )
46	45	3adant1 1079	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( C .Q B ) = ( /Q ` ( C .pQ B ) ) )
47	43, 46	breq12d 4666	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( C .Q A ) <Q ( C .Q B ) <-> ( /Q ` ( C .pQ A ) ) <Q ( /Q ` ( C .pQ B ) ) ) )
48		lterpq 9792	. . . 4 |- ( ( C .pQ A ) <pQ ( C .pQ B ) <-> ( /Q ` ( C .pQ A ) ) <Q ( /Q ` ( C .pQ B ) ) )
49	47, 48	syl6bbr 278	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( C .Q A ) <Q ( C .Q B ) <-> ( C .pQ A ) <pQ ( C .pQ B ) ) )
50	38, 40, 49	3bitr4d 300	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
51	2, 3, 4, 50	ndmovord 6824	1 |- ( C e. Q. -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   <N clti 9669   .pQ cmpq 9671   <pQ cltpq 9672   Q.cnq 9674   /Qcerq 9676   .Q cmq 9678   <Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-mq 9737  df-1nq 9738  df-ltnq 9740

This theorem is referenced by:  ltaddnq  9796  ltrnq  9801  addclprlem1  9838  mulclprlem  9841  mulclpr  9842  distrlem4pr  9848  1idpr  9851  prlem934  9855  prlem936  9869  reclem3pr  9871  reclem4pr  9872