Theorem mulnqprl 6666

Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)

Assertion

Ref	Expression
mulnqprl	|- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) )

Proof of Theorem mulnqprl

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

Step	Hyp	Ref	Expression
1		ltmnqg 6499	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. /\ w e. Q. ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
2	1	adantl 262	. . . . . 6 |- ( ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) /\ ( y e. Q. /\ z e. Q. /\ w e. Q. ) ) -> ( y <Q z <-> ( w .Q y ) <Q ( w .Q z ) ) )
3		simpr 103	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> X e. Q. )
4		prop 6573	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 6579	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
6	4, 5	sylan 267	. . . . . . . 8 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> G e. Q. )
7	6	ad2antrr 457	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> G e. Q. )
8		prop 6573	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
9		elprnql 6579	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
10	8, 9	sylan 267	. . . . . . . 8 |- ( ( B e. P. /\ H e. ( 1st ` B ) ) -> H e. Q. )
11	10	ad2antlr 458	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> H e. Q. )
12		mulclnq 6474	. . . . . . 7 |- ( ( G e. Q. /\ H e. Q. ) -> ( G .Q H ) e. Q. )
13	7, 11, 12	syl2anc 391	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( G .Q H ) e. Q. )
14		recclnq 6490	. . . . . . 7 |- ( H e. Q. -> ( *Q ` H ) e. Q. )
15	11, 14	syl 14	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( *Q ` H ) e. Q. )
16		mulcomnqg 6481	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) = ( z .Q y ) )
17	16	adantl 262	. . . . . 6 |- ( ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) /\ ( y e. Q. /\ z e. Q. ) ) -> ( y .Q z ) = ( z .Q y ) )
18	2, 3, 13, 15, 17	caovord2d 5670	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) <-> ( X .Q ( *Q ` H ) ) <Q ( ( G .Q H ) .Q ( *Q ` H ) ) ) )
19		mulassnqg 6482	. . . . . . . 8 |- ( ( G e. Q. /\ H e. Q. /\ ( *Q ` H ) e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = ( G .Q ( H .Q ( *Q ` H ) ) ) )
20	7, 11, 15, 19	syl3anc 1135	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = ( G .Q ( H .Q ( *Q ` H ) ) ) )
21		recidnq 6491	. . . . . . . . 9 |- ( H e. Q. -> ( H .Q ( *Q ` H ) ) = 1Q )
22	21	oveq2d 5528	. . . . . . . 8 |- ( H e. Q. -> ( G .Q ( H .Q ( *Q ` H ) ) ) = ( G .Q 1Q ) )
23	11, 22	syl 14	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( G .Q ( H .Q ( *Q ` H ) ) ) = ( G .Q 1Q ) )
24		mulidnq 6487	. . . . . . . 8 |- ( G e. Q. -> ( G .Q 1Q ) = G )
25	7, 24	syl 14	. . . . . . 7 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( G .Q 1Q ) = G )
26	20, 23, 25	3eqtrd 2076	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( G .Q H ) .Q ( *Q ` H ) ) = G )
27	26	breq2d 3776	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) <Q ( ( G .Q H ) .Q ( *Q ` H ) ) <-> ( X .Q ( *Q ` H ) ) <Q G ) )
28	18, 27	bitrd 177	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) <-> ( X .Q ( *Q ` H ) ) <Q G ) )
29		prcdnql 6582	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ G e. ( 1st ` A ) ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
30	4, 29	sylan 267	. . . . 5 |- ( ( A e. P. /\ G e. ( 1st ` A ) ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
31	30	ad2antrr 457	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) <Q G -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
32	28, 31	sylbid 139	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) ) )
33		df-imp 6567	. . . . . . . . 9 |- .P. = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y .Q z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y .Q z ) ) } >. )
34		mulclnq 6474	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
35	33, 34	genpprecll 6612	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) /\ H e. ( 1st ` B ) ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
36	35	exp4b 349	. . . . . . 7 |- ( A e. P. -> ( B e. P. -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( H e. ( 1st ` B ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) ) ) )
37	36	com34 77	. . . . . 6 |- ( A e. P. -> ( B e. P. -> ( H e. ( 1st ` B ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) ) ) )
38	37	imp32 244	. . . . 5 |- ( ( A e. P. /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
39	38	adantlr 446	. . . 4 |- ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
40	39	adantr 261	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) e. ( 1st ` A ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
41	32, 40	syld 40	. 2 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) ) )
42		mulassnqg 6482	. . . . 5 |- ( ( X e. Q. /\ ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = ( X .Q ( ( *Q ` H ) .Q H ) ) )
43	3, 15, 11, 42	syl3anc 1135	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = ( X .Q ( ( *Q ` H ) .Q H ) ) )
44		mulcomnqg 6481	. . . . . . 7 |- ( ( ( *Q ` H ) e. Q. /\ H e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
45	15, 11, 44	syl2anc 391	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` H ) .Q H ) = ( H .Q ( *Q ` H ) ) )
46	11, 21	syl 14	. . . . . 6 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( H .Q ( *Q ` H ) ) = 1Q )
47	45, 46	eqtrd 2072	. . . . 5 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( *Q ` H ) .Q H ) = 1Q )
48	47	oveq2d 5528	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q ( ( *Q ` H ) .Q H ) ) = ( X .Q 1Q ) )
49		mulidnq 6487	. . . . 5 |- ( X e. Q. -> ( X .Q 1Q ) = X )
50	49	adantl 262	. . . 4 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X .Q 1Q ) = X )
51	43, 48, 50	3eqtrd 2076	. . 3 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( X .Q ( *Q ` H ) ) .Q H ) = X )
52	51	eleq1d 2106	. 2 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( ( ( X .Q ( *Q ` H ) ) .Q H ) e. ( 1st ` ( A .P. B ) ) <-> X e. ( 1st ` ( A .P. B ) ) ) )
53	41, 52	sylibd 138	1 |- ( ( ( ( A e. P. /\ G e. ( 1st ` A ) ) /\ ( B e. P. /\ H e. ( 1st ` B ) ) ) /\ X e. Q. ) -> ( X <Q ( G .Q H ) -> X e. ( 1st ` ( A .P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   1Qc1q 6379   .Q cmq 6381   *Qcrq 6382   <Q cltq 6383   P.cnp 6389   .P. cmp 6392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-imp 6567

This theorem is referenced by:  mullocprlem  6668  mulclpr  6670